During Shabbat and Yom Tov, the Shmoneh Esreh consists of a total of 7 blessing. The beginning and ending 3 are the same as during weekdays.
On Shabbat, the middle blessing text is different for each service (Arvit Lel Shabbat, Shacharit and Mincha on Yom Shabbat. I have excluded Musaph as that is always different than the others.)
On Yom Tov, the middle paragraph is the same for all 3 services.
Why does Shabbat warrant a different middle section but Yom Tov does not?
The idea of Shabbos originated at Ma'asei Bereishis. (Creation). The Jews accepted the Shabbos 2448 years later.
The Shmoneh Esrei (actually, the Amidah) of each tefillah represents a different approach to Shabbos.
A. Ma'ariv refers to haShem's Creation; therefore, it starts "Atta Kidashta" this is like a heading. then we go on to say
These 3 phrases are direct references to the pesukim that are quoted immediately afterwards:
B. Shacharis refers to our acceptance of the Mitzva of Shabbos. And so the focus is on Moshe, who received the "Gift of his portion" and his receiving the two Luchos, (tablets) on which the Shabbos was one of the Dibros carved thereon.
C. Mincha refers to our Dedication to the Holiness of the Shabbos, recognizing that Our Shemirah (Observing) of Shabbos is tied inextricably to Kiddush Shem Shamayim (Sanctification of HaShem's Name)
In short:
The first stresses HaShem's input.
The second Amidah stresses our input.
As for YomTov, all Amidahs are essentially identical (except for references to Shabbos or Motzaei Shabbos) because the theme in all of them is Yetzias Mitrayim (the Exodus from Egypt).
This beureihatefillah.com article mentions that the concept of having a different middle evolved much later. Siddur Rav Amram Ga'on mentions that, originally, the middle section was alike, similar to the Yom Tov davening.
Machzor Vitri 162 says:
You may be wondering why we change the middle Bracha of Shemona Esrei for the evening, the morning and the afternoon services and we do not recite the same Bracha in each service as we do on Yom Tov when we recite the Bracha of Ata Bichartanu in each Tefila. I heard from one Rabbi that the three versions of Shemona Esrei commemorate three different Shabbosim. Ata Kidashta was composed to commemorate Shabbos Berieshis which G-d sanctified and which resulted from the six days of creation. Yismach Moshe was composed to commemorate the Shabbos of Mount Sinai when the Torah was given on Shabbos early in the day as it is written in the chapter of Gemara known as Rabbi Akiva that according to both opinions the Torah was given in the morning. The fact that we recite Tikanta Shabbos for Mussaf on Shabbos is not surprising because even on holidays we change the wording of Mussaf Shemona Esrei from the Shemona Esrei of Tefilas Shacharis, Mincha and Maariv. Ata Echad was composed to commemorate the everlasting Shabbos which will begin at the time of the Moshiach as it is written: on that day G-d will be one and His name will be one and the Jewish people will be one nation as it is written: Your nation all righteous men, will inherit the world forever.
I found it amazing that this question has come up 3 times in the last 3 weeks. I went to a shul Seudah this past week and the MC, Mr Luchanski, brought up the question and he answered similarly for the first two tefillos. However, he phrased it that maariv reminisces about the Creation, Shacharis relates to Kabbalas HaTorah, and Mincha refers to Yemos haMashiach (when the Messiah will come). His proof was that Mincha is in the future tense. He may have also mentioned that "shimcha echad"(your name is One) will only occur in the future. There are difficulties with that (1) Shacharis is also in the future tense; and (2) "Shimcha echad" seems to be in the present tense. The first difficulty may parallel elsewhere where Moshe is referred to in the future, "Oz yashir Moshe," (then Moshe WILL sing) Exodus 15/1 according to the Midrash, a "proof" that haShem will resurrect the dead. Of course that would mean that Chazal chose to mirror the idea by using the future tense. The Rashi and the Ramban both seek to explain that that is not the simple P'shat. And the Ramban states "it is the way of the [Scriptural] language to speak in the future instead of the past."
As for "Shimcha echad," the idea that that refers specifically to the future was hinted in the prophecy of Ovadiah, read for last week's Haftarah. Saviors will...judge the mountain of Eisav, [then] the Kingdom will belong to HaShem."
We quote this verse daily, after saying "Oz yashir Moshe" and we add a verse from Zecharia, (14/9) "On that day, haShem will be One and His Name (shimcha) will be One." One could say that haShem is One and His name is One, even now, by virtue of the Jewish people accepting Him! But, when Eisav/Amalek is finally defeated and haShem will be Accepted as Ruler over the Entire World, then his Oneness And His One Name, will be spread over and accepted by the whole world. [I think that is what the Malbim says. See his commentary on 14/9; 13/1; also on Yechezkel, that water will flow from the Holy of Holies to flow to the entire world, which he explains is, besides any literal p'shat, a parable for knowledge
The Maharal in Derech Chaim, on the Hakdama of Kol Yisroel Yesh Lahem Chelek, writes that the three Tefillos are according to the three levels of Shabbos which are:
