I know both the Rambam and Ramban listed all of the 613 mitzvos, and that they differ in some (I think like 16ish?)
What are the ones upon which they differ? If possible please include why they differ in specific instances.
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The following is just my understanding of the sources, and if there are any mistakes (ex: in the Ramban's reasons to disagree with the Rambam, the titles given to mitzvos, etc.), please point them out to me. I used the Mishneh LaMelech's Derech Mitzvosecha § 1, this article, and the Ramban's summary at the end of his Hasagos to help with this list.
In summary: The Ramban removes 56 mitzvos (32 + 24) , and adds 63 mitzvos (32 + 31, see fn. 4).
If you would like to see the list of mitzvos in a more visually appealing format, click here to see a Beta version of a website I'm working on (Desktop only).
The sefer Seder HaMitzvos also writes a summary of the differences between the Rambam and the Ramban. Note he says the Ramban removed 30 positive mitzvos and not 32. For semantic reasons he didn't mention removal #7 and #29, but there's no disagreement.
(Lettered entries aren't removed from Ramban's list, rather split in two)
2) [#7] מצות על כל הצריך לישבע שישבע בשם השם - Anyone who has to swear to only swear in the name of Hashem. Because the verse is giving permisison, not commanding.
3) [#63] מצות מעשה העולה - The Olah offering procedure. Because as per Shoresh #12, this is just a detail of the mitzvah to bring offerings.
4) [#64] מצות מעשה החטאת - The Chatas offering procedure. Because of #3.
5) [#65] מצות מעשה האשם - The Asham offering procedure. Because of #3.
6) [#66] מצות מעשה זבח השלמים - The Shelamim offering procedure. Because of #3.
7) [#67] מצות קרבן מנחה - The Mincha offering. Because of #3.
8) [#85] מצות להטפל בהבאת הקרבן מחוצה לארץ לבית הבחירה - To take care of the offerings as they are brought from outside the land to the Temple. Because anyways we need to bring offerings to the Temple, regardless of where they are, so this isn't a new command.
9) [#95] מצות דין הפרת נדרים - Nullification of vows. Because it is really a shelilah, to do all that which leaves our mouths and not to profane our words except through the annulment of Beis Din, a father, or a husband (see Shoresh #8).
11) [#97] מצות טומאת שמנה שרצים - The concept of tumah of the eight sheratzim. Because of #10.
12) [#98] מצות עניין טומאת אוכלים - The concept of tumah of food. Because of #10.
13) [#99] מצות עניין טומאת נדה שטמאה ומטמאה - The concept of tumah of a niddah. Because of #10.
14) [#100] מצות עניין טומאת יולדת - The concept of tumah of a woman who gave birth. Because of #10.
15) [#101] מצות עניין טומאת מצורע - The concept of tumah of a metzora. Because of #10.
16) [#102] מצות עניין נגעי בגדים - The concept of tumah of tzaraas on a garment. Because of #10.
17) [#103] מצות עניין טומאת בית שיהיה בו נגע - The concept of tumah of tzaraas on a building. Because of #10.
18) [#104] מצות עניין טומאת זב להיות טמא ומטמא - The concept of tumah of a zav. Because of #10.
19) [#105] מצות עניין טומאת שכבת זרע שהוא טמא ומטמא - The concept of tumah of someone who had a seminal emission. Because of #10.
20) [#106] מצות עניין טומאת זבה שטמאה ומטמאה - The concept of tumah of a zavah. Because of #10.
21) [#107] מצות טומאה של מת - The concept of tumah of a corpse. Because of #10.
22) [#108] מצות מי נדה שמטמאין אדם טהור ומטהרין אדם טמא מטומאת מת בלבד - The concept of the water of a red heifer which purifies and contaminates. Because of #10.
g) [#130] מצות מעשר עני תחת מעשר שני בשנה שלישית - To separate Maaser Ani instead of Maaser Sheni in the third year. The Ramban splits this mitzvah into two, this one being about separating Maaser Ani. The other is below addition #25.
23) [#142] מצות לנגוש את הנכרי - To pressure a non Jew to repay a debt. Because this is a lav habah miklal aseh and since there is already a lav counted, this shouldn't be as well.
24) [#149] מצות בדיקת סימני בהמה וחיה - To check the kosher signs of domesticated and wild animals. Because this is a lav habah miklal aseh and since there is already a lav counted, this shouldn't be as well.
25) [#150] מצות בדיקת סימני העוף - To check the kosher signs of birds. Because of #24.
26) [#151] מצות בדיקת סימני חגבים - To check the kosher signs of grasshoppers. Because of #24.
27) [#152] מצות בדיקת סימני דגים - To check the kosher signs of fish. Because of #24.
28) [#198] מצות הלואה לנכרי ברבית אם יצטרך ללוות מה שאינו כן בישראל - To lend to a non Jew with interest if he needs a loan, unlike to a Jew. Because of #23.
29) [#226] מצות בית דין להרוג בסייף המחוייב - For Beis Din to kill those liable to decapitation. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
30) [#227] מצות בית דין להרוג בחנק המחוייב - For Beis Din to kill those liable to strangulation. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
31) [#228] מצות שישרפו מי שיתחייב שריפה - For Beis Din to kill those liable to death by fire. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
32) [#229] מצות על בית דין לסקול באבנים מי שיתחייב סקילה - For Beis Din to kill those liable to death by stoning. Because there's a single mitzvah to kill those sentenced to death, and the different ways are details.
1) [#5] שלא להשתחוות לעבודה זרה - Not to bow to idols. Because this is included in Rambam's #1, not to believe in idols.
2) [#6] שלא לעבוד עבודה זרה במה שדרכה להעבד - Not to worship idols in their normal mode of service. Because this is included in Rambam's #1, not to believe in idols.
3) [#28] שלא לשמע ממתנבא בשם עבודה זרה - Not to listen to someone who prophecizes in the name of an idol. Because this is just another verse prohibiting idol worship.
a) [#53] שלא ישא עמוני ומואבי בת ישראל - An Ammonite or Moabite cannot marry a Jewish woman. The Ramban splits this mitzvah into two, this one being about an Ammonite. The other is below, addition #4.
5) [#69] שלא יכנס בעל מום בהיכל - A blemished Kohen shall not enter the Temple Heichal. Because "he shouldn't enter" refers to doing the Temple service, which is already in mitzvah #70.
c) [#100] שלא להקריב אתנן זונה ומחיר כלב - Not to offer a harlot's wage or the exchange for a dog. The Ramban splits this into two, this one being about a harlot's wage. The other is below, addition #9.
d) [#125] שלא לאכול הפסח נא ומבושל - Not to eat the Pesach offering raw or boiled. The Ramban splits this into two, this one being about a eating the Pesach offering raw. The other is below, addition #10.
8) [#142] שלא לאכול מעשר שני של תירוש חוץ לירושלם - Not to eat Maaser Sheni of wine outside Jerusalem. The Rambam counts these as three mitzvos (#141-143), but the Ramban combines them all into one.
9) [#143] שלא לאכול מעשר שני של יצהר חוץ לירושלם - Not to eat Maaser Sheni of oil outside Jerusalem. The Rambam counts these as three mitzvos (#141-143), but the Ramban combines them all into one,
10) [#150] שלא לאכול מעשר שני בטומאה - Not to eat Maaser Sheni while impure. The Ramban holds that this is just another case of shelilus plus it's a lav habah miklal aseh, which the Rambam says not to count.
11) [#151] שלא לאכול מעשר שנע באנינות - Not to eat Maaser Sheni as an Onen. The Ramban holds that this is just another case of shelilus plus it's a lav habah miklal aseh, which the Rambam says not to count.
12) [#152] שלא להוציא דמי מעשר שני בשאר דברים שאינם מאכילה ושתייה - Not to purchase with Maaser Sheni money anything besides food or drink. The Ramban holds that this is just another case of shelilus plus it's a lav habah miklal aseh, which the Rambam says not to count.
13) [#177] שלא לאכול מן השרצים המתהוים מן העפוש - Not to eat bugs that come from sweat. As per Ramban's understanding of Shoresh #9, if it's essentially one prohibition, it doesn't matter if there are many multiples of lashes.
14) [#178] שלא לאכול מיני שרצים דקים הנולדים בזרעים ובפירות - Not to eat worms that come from seeds or fruits. Because of #13.
15) [#179] שלא לאכול משרץ המים - Not to eat sea bugs. Because of #13.
16) [#190] שלא לאכול קלי מתבואה עד ביום ההוא - Not to eat roasted grain until the Omer is brought. Because of #13.
17) [#191] שלא לאכול כרמל מתבואה עד הזמן הנזכר - Not to eat fresh grain until the Omer is brought. Because of #13.
19) [#198] שלא לאכול מכל דבר שיש בו חמץ - Not to eat anything that has chametz in it. This is just an extra prohibition against chametz, so it's not worth counting again.
20) [#199] שלא לאכול חמץ אחר חצות - Not to eat chametz after noon. This is the opinion of R' Yehudah, but we hold like R' Shimon that there's no negative commandment on this.
f) [#290] שלא לחתוך הדין באומד הדעת - Not to decide someone's fate just with estimation. The Ramban splits this mitzvah into two, this one being about killing someone without proper testimony (see Megillas Esther ad. loc.). The other is below, addition #22.
21) [#294] שלא לענוש האנוס - Not to punish someone coerced. The Ramban holds that this is just another case of shelilus, that they aren't punished, not that it's a command.
22) [#307] שלא לגלח שער הנתק - Not to shave the hair of a nesek affliction. This is the same as #308, and no need to count it as two mitzvos.
23) [#318] שלא לקלל אב ואם - Not to curse one's father or mother. This is the same prohibition against cursing any Jew, despite having a different punishment.
24) [#319] שלא להכות אב ואם - Not to injure one's father or mother. This is the same prohibition against hurting any Jew, despite having a different punishment.
5) שלא לשחוט קרבנות על מנת לאכלם או לעשות עבודתם חוץ למקומם או לזמנם - Not to slaughter offerings with intention to eat them or perform their service outside of their location or time (source, description).
6) שלא לאכול קרבנות ששחטן על מנת לאכלן או להקריבן חוץ למקומן - Not to eat offerings which were slaughtered with intention to eat them or perform their service outside of their location (source, description).
1 There seems to be a typo in Derech Mitzvosecha, which says the Ramban split mitzvah #247 into two.
2 The following mitzvah in Shichchas HaAsin LeDaas HaRambam #6, has the mitzvah of מצות לאכול מן האילנות שבגבולה כל ימי המצור ולא לכרות אותם - To enjoy the fruits of the trees surrounding a city under siege and not to destroy them (source). The Mishneh LaMelech claims that the Ramban listed it only according to the opinion of the Rambam, who himself should have listed it. However, the Ramban himself doesn't hold it should be included, for the same reason as #24 above. This explains why the Ramban at the conclusion of his Hasagos says there are only 16 mitvos that the Rambam forgot, when he himself listed 17.
3 At the end of Ramban's Hasagos, he says that he's unsure if he should list the positive mitzvah for a King or Shofet to ask the Urim VeTumim if the people should go out to war. He says if it shouldn't be counted, then he'll count the mitzvah for a Sage to rule while not intoxicated. He leaves it as an unresolved question, but in his Hasagos to negative mitzvah #73 he pretty confidently counts the latter as a mitzvah, so I wrote above only that one. However, at the end of the Zohar HaRakiah he writes that we cannot rely on what the Ramban writes in his Hasagos; only on what he writes at the end when he sums up the mitzvos. This is also brought in the sefer Seder Hamitzvos.
4 The Ramban in his summary at the end of his Hasagos writes that the Rambam is missing seven negative mitzvos. Apparently he had a different version of Sefer HaMitzvos than we have. The Zohar HaRakiya tries to figure out which were missing, and feels he discovered five. However, Rav Chaim Heller (Yevul HaYovlos pg. 82) disagrees, and feels there was simply a copyist error at fault. He therefore says it's a waste of time trying to figure out which mitzvos were missing from his version. Regardless, the Ramban therefore added 7 more to the 24 he removed.
The list of differences can be found in an appendix to the Minchas Chinuch.
Mitzvah 1: Positive mitzvah of eating maaser sheni.
Mitzvah 2: Positive mitzvah of not eating impure trumah
Mitzvah 3: Positive mitzvah of not selling shvi'is fruits
Mitzvah 4: Positive mitzvah of settling Eretz Yisrael
Mitzvah 5: Positive mitzvah of leaving a way for the enemy to retreat
Mitzvah 6: Positive mitzvah to eat fruit trees when sieging a city
Mitzvah 7: Positive mitzvah to remember Miriam verbally
Mitzvah 8: Positive mitzvah to be wholesome with G-d
Mitzvah 9: Positive mitzvah to separate trumah from the best
Mitzvah 10: Positive mitzvah for every korban to be one of the three valid species
Mitzvah 11: Positive mitzvah to offer all korbanos between the two temidin
Mitzvah 12: Positive (?) mitzvah for the Pesach not to be eaten by day
Mitzvah 13: Positive mitzvah to kill a killer
Mitzvah 14: Positive mitzvah not to benefit from clothing or a house with tzara'as
Mitzvah 15: Positive mitzvah to thank G-d
Mitzvah 16: Positive mitzvah to keep your brother alive
Mitzvah 17: Positive mitzvah to return interest
Mitzvah 1: Negative mitzvah not to forget belief in G-d
Mitzvah 2: Negative mitzvah not to forget the standing at Har Sinai
Mitzvah 3: Ngeative mitzvah not to misplace vessels in the beis hamikdash
Mitzvah 4: Negative mitzvah not to slaughter kodshim intending to eat them
Mitzvah 5: Negative mitzvah not to eat kodshim outside of their place
Mitzvah 6: Negative mitzvah not to eat the bird slaughtered for a metzora
Mitzvah 7: Negative mitzvah for levi'im not to take trumah
Mitzvah 8: Negative mitzvah not to accept shekalim from non-Jews for the purpose of korbanos tzibur
It is important to note that the Rambam and Ramban agree with each other most of the time that these are obligations, but they disagree about whether they are de'oraisa.