Gemara

Mishnah: If the Megilah was read in Adar Rishon, and the year was declared a leap year, we read it again in Adar Sheni;

The only differences between Adar Rishon and Sheni are Kri'as ha'Megilah and Matanos la'Evyonim.

Gemara (Inference): Adar Rishon and Sheni are the same regarding the four special (Maftir) portions.

Question: The Mishnah is not like any of the Tana'im in the Beraisa!

Beraisa: If the Megilah was read in Adar Rishon and the year was declared a leap year, we read it again in Adar Sheni. All Mitzvos which apply in Adar Sheni apply in Adar Rishon, except for Kri'as ha'Megilah;

R. Eliezer bar R. Yosi says, we do not read it in Adar Sheni, because all Mitzvos that apply in the second apply in the first;

R. Shimon ben Gamliel says, we also read in Adar Sheni, since all Mitzvos that apply in the second don't apply in the first.

Eulogies and fasting are forbidden on the 14th and 15th of both months.

Question: What is the difference between the first Tana and R. Shimon ben Gamliel?

Answer (Rav Papa): They argue about the four Parashiyos. The first Tana says that if they were read in Adar Rishon, one need not repeat them; R. Shimon ben Gamliel says that they must be repeated.

Summation of question (4): The Mishnah is unlike the first Tana, who says that even Matanos la'Evyonim do not apply in Adar Rishon. It is not like R. Eliezer bar R. Yosi, who says that the Megilah may be read in Adar Rishon. It is not like R. Shimon ben Gamliel, who says that even the Parashiyos must be repeated.

Answer #1: Really, the Mishnah is like the first Tana. 'Except for Kri'as ha'Megilah', includes Matanos la'Evyonim, since these depend on the reading.

Answer#2: The Mishnah is like R. Shimon ben Gamliel. It means that the only difference between the 14th of Adar Rishon and Sheni is Kri'as ha'Megilah and Matanos la'Evyonim. They are the same regarding eulogies and fasting. The Mishnah does not discuss the Parashiyos.

R. Chiya bar Avin: The Halachah follows R. Shimon ben Gamliel.

Rishonim

Rif: Regarding the 14th, the Mishnah says that the only differences between Adar Rishon and Sheni are Kri'as ha'Megilah and Matanos la'Evyonim. They are the same regarding eulogies and fasting.

Ran (DH Ela): The Gemara need not mention Seudas Purim because it says that eulogies and fasting are forbidden in both months, and the source to forbid eulogies and fasting is "Mishteh v'Simchah" (so surely the Seudah also applies in Adar Rishon). Alternatively, the Mitzvah of a Seudah is mid'Rabanan (i.e. not expounded from the verse). Nevertheless it is proper to have a bigger Seudah than normal in Adar Rishon. Mishlo'ach Manos is like Matanos la'Evyonim, which is only in Adar Sheni. Even though villagers who read early give Matanos la'Evyonim that day but do not send Mishlo'ach Manos the same day, that is because they do not eat the Seudah that day. (The Mitzvah of Mishlo'ach Manos is to send portions which can be eaten at the Seudah.)

Ran (DH Gemara): If the four Parashiyos were read in Adar Rishon they must be repeated in Adar Sheni.

Rambam (Hilchos Megilah 2:13): Eulogies and fasting are forbidden on the 14th and 15th for everyone everywhere, both in Adar Rishon and Sheni.

Rosh (1:7): The Mishnah says that the only differences between Adar Rishon and Sheni are Kri'as ha'Megilah and Matanos la'Evyonim if the year was made Me'uberes after the 14th. However, anything else that was done in Adar Rishon need not be repeated. If the year was made Me'uberes before Adar Rishon, we do not do any Mitzvos of Adar in Adar Rishon. Eulogies and fasting are the same in both, i.e. if the Megilah was read on the 14th of Adar Rishon and they were Me'aber the year that day. It is unreasonable that eulogies and fasting would be forbidden even if the Megilah was not read.

Poskim

Shulchan Aruch (OC 697:1): On the 14th and 15th of Adar Rishon we do not say Tachanun or "Ya'ancha Hash-m b'Yom Tzarah". Eulogies and fasting are forbidden, but other laws of Purim do not apply. Some permit even eulogies and fasting.

Gra (DH v'Yesh Omrim): From the Seifa (the only difference between Adar Rishon....) we inferred that Simchah and Mishteh are the same in both. The lenient opinion holds that the Seifa must refer to the Reisha, when the year was made Me'uberes after they read the Megilah. It cannot discuss when we knew in advance that the year will be Me'uberes, for we would never think that one must read the four Parashiyos in both months. The Ran rejects this; the Mishnah means, even in such a case R. Shimon ben Gamliel requires repeating the Parashiyos in Adar Sheni.

Rema: The custom is like the first opinion. Some obligate extra Simchah and Mishteh on the 14th of Adar Rishon; this is not our custom. In any case it is good to increase a bit to fulfill this opinion. "V'Tov Lev Mishteh Samid".

Beis Yosef (DH u'Pashut): Some infer that the only difference between Adar Rishon and Sheni are Kri'as ha'Megilah and Matanos la'Evyonim, but Simchah and Mishteh are the same in both. Tosfos (6b DH v'R. Eliezer) rejects this, because the Gemara infers that they are the same regarding eulogies and fasting, but not regarding Simchah and Mishteh. SMaK says that even though we increase Simchah and Mishteh on the 14th and 15th of Adar Sheni, in Adar Rishon we do so only on the 14th. This is because the Gemara says that the only difference between the 14th of Adar Rishon... It does not mention the 15th. However, perhaps it only mentions the primary day. There is no Isur Melachah in Adar Rishon. Even in Adar Sheni the Isur depends on custom. Nowadays the custom is not to increase Simchah and Mishteh in Adar Rishon. Nevertheless, we do not say Tachanun or "Ya'ancha Hash-m b'Yom Tzarah" because they are days of a miracle and salvation.

Mishbetzos Zahav (1): If someone vowed to fast on Purim Katan it takes effect, for it is a mere custom not to fast. However, if he vowed to fast on Mondays and Thursdays, presumably he did not intend to forbid Purim Katan.

Eshel Avraham (2): The Rema brings "V'Tov Lev Mishteh Samid" to suggest increasing Simchah also on the 14th of Adar Rishon. The Magen Avraham (2) explains that this refers to Simchah in Avodas Hash-m.