Is there any significance in Judaism attached to the day someone was born (aside for the 13th year)? Is there anything wrong with celebrating one's birthday? And is it more correct to celebrate the "hebrew" date or the gregorian date?
There are definitely some specially celebratory ones other than age 13, such as when one surpasses the deadline for kares, as Rav Yosef did when he turned 60 and threw himself a party (Mo'ed Katan 28). The Kaf Hachayim, cited in this article by Rav Ari Enkin, also quotes sources for age 70 being an appropriate birthday to recite birkas shehechiyanu (presumably due to its identification with the human lifespan). That same article presents plenty of sources for the significance of the birthdays of great people in our history, both from practical and metaphysical standpoints.
The assumption in all of the above is that the birthday of significance is the Jewish one. For example, (as you noted) it is what determines the date that one reaches legal adulthood, and all other considerations in halacha.
Sichos in English has a list of birthday customs suggested by the last Lubavitcher Rebbe.
A birthday is a time to reflect and to accept upon oneself to improve in one's ways. The correct day to celebrate is the Hebrew date. It is significant as we see in Pirkei Avos Ben Esrim, Ben Shloshim, etc.
Rav Aviner addresses this question as follows
Q: Is there a source for celebrating one's birthday?
A: There is no early source besides the Torah's mention of Pharoah's birthday. There is no mention in the Mishnah, Gemara, Rambam or Shulchan Aruch. The Aderet harshly opposed birthday celebrations. The book "Nefesh David". And some authorities opposed it on account of "Chukot Ha-Goyim" – non-Jewish customs. But one may permit it since it is a custom based on a rational reason. The Chazon Ish did not celebrate his birthday in any way (in the name of Ha-Rav Chaim Kanievski. Segulot Raboteinu p. 350). And Rabbenu Ha-Rav Tzvi Yehudah said that there is some value in celebrating one's birthday on condition that one does a Cheshbon Nefesh (taking an account of one's spiritual state) on that day.
There is a notion that someone celebrating a birthday is a Chiyuv for getting an Aliyah to the Torah. It comes really low on the list, but it is there.
According to the Ben Ish Chai it is important. But even more important is one's bris day. He writes in shana alef parshas Re'eh ois 17 that people are noheig to make their birthday into a yom tov, and it is a good siman, he adds that he too did it. He goes on to describe the bris-day minhag.
It doesn't seem like there's any real significance to an ordinary birthday. But it doesn't seem like there's anything wrong with marking another lap around the sun. For Jewish matters, the hebrew birthday is what counts, so I assume that would be the main day. However, see this post: Secular Birthdays.
Hirhurim also discussed birthdays here: Birthdays.
Though over here, R.Enkin found some scattered sources that might support it: Birthdays.
I've heard argued that the secular date should be celebrated because the only source in tanach that we have of birthdays is Pharoh celebrating his birthday in Gen 40:20 There is no reference in tanach to a hebrew birthday. Hence the secular date should be celebrated.
R. Joseph Hayim of Baghdad talks about birthdays in his commentary to Masechet Berachot. He writes that birthdays are a time of special "luck" and success for the person, and that it is therefore customary for everyone to make their birthdays into a holiday for themselves:
Ben Yehoyada Berachot 28a
ההוא יומא בר תמני סרי שני הוה. נראה לי בס"ד דקדק לומר ההוא יומא ולא אמר ההוא זמנא דאפשר אותו היום היה יום הלידה שלו והיה י"ח שנה שלמים ולכך הצליח לעשות לו נס זה דידוע שיום הלידה יהיה המזל של האדם חזק בו ומוצלח על כן נוהגים שכל אדם יעשה יום הלידה יום טוב לעצמו
To the first question, according to the Chida in Chomat Anach (Iyov 3:3) there is kabbalistic significance associated with one's birthday.
אפשר במ"ש המקובלים שהיום שנולד בו האדם מזלו בריא וחזק בכל שנה
to paraphrase, a person's spiritual influence is strengthened on her/his birthday each year. Perhaps such a spiritual strengthening is worthy of celebration?
My assumption, though it is not stated explicitly, is that this spiritual influence would be tied to the Hebrew date of birth since it is related to a person's mazal