(Beraisa #1): If she went to a Chacham and he permitted her [vow], she is Mekudeshes. If she went to a doctor and he cured her [of her Mum], she is not Mekudeshes.

Question: What is the difference between a Chacham and a doctor?

Answer: A Chacham uproots the vow from its source (retroactively). A doctor cures only from now and onwards.

(Beraisa #2): Some teach that even if she went to a Chacham and he permitted her, she is not Mekudeshes.

Beraisa (#1) is like R. Elazar (who holds that when a vow is uprooted retroactively, also any adamancy about it is uprooted. Rabanan, i.e. Beraisa #2, hold that at the time of Kidushin he did not want a woman with vows, so the Kidushin was invalid, even if later the vows are permitted retroactively.)

(Mishnah - R. Elazar): They forbade [remarrying a wife that he divorced due to] this (a vow that her husband could annul by himself) only due to this (a vow that needs a Chacham to permit it)!

Question: What is R. Elazar's reason?

According to letter of the law, even a vow that needs Chakiras Chacham, he returns her, for a Chacham uproots the vow from the beginning (it is as if she never vowed, so the Get was mistaken; she is still married. Chachamim enacted that he may not return her, so he will divorce unconditionally, lest she marry someone else, permit her vow, and transgress Eshes Ish and bear Mamzerim);

Why did they forbid [returning her in the case of] a vow that does not need Chakiras Chacham? (He did not divorce due to it, for he could have annulled it. There is no concern!) It is due to [confusion with] a vow that needs Chakiras Chacham. (If he may remarry her in this case, people will come to remarry even in the former case. Even though this is like a decree due to a decree, Chachamim decreed, due to the stringency of Arayos and Mamzerus.)

Some teach that she is permitted to marry without a Get, and some teach that she is forbidden to marry without a Get;

Assumption: They wanted to say that the one who permits to marry without a Get is R. Lazar (he says Einah Mekudeshes only regarding Mumim, which cannot be fixed retroactively), and the one who forbids to marry without a Get is Rabanan.

Note: Perhaps Rabanan are unsure whether or not adamancy is uprooted retroactively, and 'she is not Mekudeshes' is a stringency, to be concerned if another was Mekadesh her, or her husband was Mekadesh her sister, but they are not lenient to let her remarry without a Get. Alternatively, they are stringent lest people think like R. Elazar, and say that she is Mekudeshes to the first, and that her children from her new husband are Mamzerim.)

Rejection: Both are like R. Elazar (before she permits her vow). The one who permits to marry without a Get holds that since she knows that if she will go to a Chacham, and he will permit her vow [her Kidushin takes effect retroactively, and she is Chayav Misah for Bi'ah with her new husband and their children are Mamzerim]. Therefore [surely she will not go, and] she is permitted without a Get;

The one who forbids to marry without a Get, this is lest she go to a Chacham, he will permit her vow, and her Kidushin takes effect retroactively, and it turns out her children are Mamzerim. Therefore, she is forbidden to marry without a Get.

Note: I did not see anyone explain like this, but this Perush explains also the Sugya in Gitin without changing the text or introducing concerns not mentioned in either Sugya (lest women wantonly vow, or one does not want the shame of going to Beis Din to permit vows...), and requires only one small textual change here, after equating our text to that of Gitin - PF.)