Gemara

Beraisa: Chachamim did not require one to mention dew and wind. If he wants to mention them he may.

R. Chanina: One is not obligated because they are never withheld.

3b - R. Chanina: Therefore, if in summer one mentioned wind he does not go back. If he mentioned rain he must go back;

In winter if one omitted (mention of) wind he does not go back; if he omitted rain he must go back. Even if he said '(Hash-m) removes wind and casts away dew', he does not go back.

Berachos 29a - R. Tanchum: If one forgot to mention rain in Techiyas ha'Mesim (the second Berachah), he must go back; if one forgot to request rain in Birkas ha'Shanim, he does not go back, for he may say it in Shome'a Tefilah.

Question (Beraisa): If one forgot to mention rain in Techiyas ha'Mesim or to request rain in Birkas ha'Shanim, he must go back.

Answer #1: One who prays with a Minyan does not go back; one who prays alone must go back.

Objection: Presumably, the distinction is that with a Minyan he does not go back because he will hear it from the Shali'ach Tzibur. But R. Tanchum's reason was because he may say it in Shome'a Tefilah!

Answer #2: Rather, R. Tanchum also discusses one who prays alone;

If one remembers before Shome'a Tefilah he does not go back, for he may say it in Shome'a Tefilah. If he remembers after Shome'a Tefilah, he must go back.

Rishonim

Rif: In winter if one omitted mentioning wind he does not go back. If he omitted mentioning rain he must go back.

Yerushalmi - R. Ze'ira: If one mentioned dew in winter he does not go back.

Question: A Beraisa says that one goes back if he omitted requesting or mentioning rain!

Answer: He goes back if he did not mention rain or dew.

Question (Ran DH Hayah): The Bavli says that one who did not mention rain in winter goes back. Presumably, we discuss one who mentioned dew, like he usually does in summer. Also, R. Chanina taught that in winter if one omitted wind he does not go back. He did not teach a bigger Chidush, that even one who omitted rain need not go back (if he mentioned dew), for he holds that he must always go back for this. Why did the Rif bring the Yerushalmi, which argues? Nevertheless, the Ra'avad says that since the Rif brought it, we rely on it. The Ramban says that we may not add to it, i.e. to say that requesting dew should suffice in place of requesting rain in winter.

The Rif brings the Gemara in Berachos.

Inference (Rif): If one remembers that he forgot to mention rain, even before Shome'a Tefilah, he must return to the beginning.

Ran (DH Bavli): One cannot compensate for this in Shome'a Tefilah because the first three Berachos are praise, and Shome'a Tefilah is request. Tosfos relies on the Yerushalmi, which says that one can compensate in Shome'a Tefilah also for omitting mention. However, if the Bavli agreed it would have taught this together with compensation for not requesting in Birkas ha'Shanim. The Beraisa in Berachos said that if one forgot to mention or request rain he must go back. The Gemara said that this is if he remembers after Shome'a Tefilah; if he remembers beforehand, he says it in Shome'a Tefilah. This refers only to request, but one who did not mention rain must go back in any case.

R. Yonah (19b DH Aval): I found an accurate text of the Bavli which says that one need not go back if he forgot to mention or request rain, because he can say it in Shome'a Tefilah. Presumably, Tosfos had this text.

Rambam (Hilchos Tefilah 10:8): If in winter someone erred and did not mention rain or dew he must go back to the beginning. If he mentioned dew he does not go back.

Question: Dew is never withheld, so we are not concerned if one omitted it in summer. Why does mention of it suffice (b'Di'eved) in place of mention of rain in winter?

Answer (Kesef Mishneh): In winter the land needs water. Mention of dew alludes to Hash-m's Gevurah, that He provides water for the land. Alternatively, dew of Berachah can be withheld, so mention of it in winter is like mention of dew of Berachah, so it is as if he mentioned rain. Most dew is not of Berachah, which is never withheld, so he need not return for omitting dew in summer.

Rosh (1:1): In winter if one omitted mentioning wind he does not go back. If he omitted rain he must go back. Avi ha'Ezri says that he does not go back to the beginning of the Berachah. Rather, he says Meshiv ha'Ru'ach u'Morid ha'Geshem Mechalkel Chayim... This is unlike one who mistakenly mentioned rain, for in that case returning to Mechalkel Chayim does not negate what he said. It seems to me that if one realized that he omitted rain after finishing the Berachah he starts Shemoneh Esre again, but if he remembered in the middle of the Berachah he says Meshiv ha'Ru'ach u'Morid ha'Geshem there and continues. This is because Chachamim did not fix a place in the Berachah where Meshiv ha'Ru'ach u'Morid ha'Geshem must be said. The custom is to say it before Mechalkel Chayim because rain provides food.

Avi ha'Ezri says that if one realized that he omitted Meshiv ha'Ru'ach u'Morid ha'Geshem after finishing Mechaye Mesim but before beginning Atah Kadosh he may say it there. This is like the Gemara says regarding one who omitted Ya'aleh v'Yavo or Retzei in Birkas ha'Mazon after Bonei Yerushalayim but before beginning the next Berachah.

The Rosh (Berachos 4:14) brings the Gemara in Berachos.

Rosh (ibid.): The Yerushalmi says that one can compensate in Shome'a Tefilah for omitting request, even though it is b'Dochak (greatly needed); all the more so one can compensate in Shome'a Tefilah for omitting mention, which is not b'Dochak (it is mere praise). The Bavli mentions compensation in Shome'a Tefilah only for request. Perhaps the Bavli argues. Alternatively, it holds that the Kal va'Chomer is so obvious that it need not be said. However, most texts of the Bavli say that one returns for omitting mention, but not for omitting request, because one can request in Shome'a Tefilah.

Poskim

Shulchan Aruch (OC 114:5): If in winter one omitted Morid ha'Geshem and did not mention dew he must go back. If he mentioned dew he does not go back.

Mishnah Berurah (25): This applies even to the first Tefilah in which we start to mention rain, i.e. Musaf of Shemini Atzeres.

Shulchan Aruch (6): If one omitted Morid ha'Geshem, finished the Berachah and started the next Berachah he must return to the beginning. If he did not yet finish the Berachah he says it where he remembered.

Mishnah Berurah (29): If one remembered after 'v'Ne'eman Atah...' he must repeat this after Meshiv ha'Ru'ach u'Morid ha'Geshem, for the closing Berachah must resemble the last words preceding it.

Mishnah Berurah (30): If he said 'Baruch Atah Hashem' he should finish 'Mechaye ha'Mesim' and say Meshiv ha'Ru'ach u'Morid ha'Geshem afterwards.

Shulchan Aruch (ibid.): Even if he finished the Berachah, if he did not start the next Berachah he does not go back. He says Meshiv ha'Ru'ach u'Morid ha'Geshem there, without a closing Berachah.

Mishnah Berurah (32) and Bi'ur Halachah (DH b'Lo): Many Rishonim say that once one finishes a Berachah it is as if he started the next Berachah, so he may not correct any omission. If one remembered after saying Baruch Atah Hash-m he should conclude "Lamdeni Chukecha" and correct his omission before finishing the Berachah, to fulfill all opinions.

Kaf ha'Chayim (42): If he said Baruch Atah Hash-m he should conclude 'Mechaye ha'Mesim' and say Meshiv ha'Ru'ach u'Morid ha'Geshem within Toch Kedai Dibur.

Rema: The first three Berachos are considered like one.

Source (Gra DH Sheloshah): We learn from Berachos 34a.

Mishnah Berurah (33): This is because all three are one matter, to praise Hash-m before requesting. Likewise, the last three are one matter, like a slave thanking his master for giving him food.

Rema: Whenever one erred he must return to the beginning, whether he is an individual or a (Shali'ach) Tzibur.