It is a matter of major halachik contention among the rabbinic authorities as to the status of Adar Rishon.
Many ask, is it considered to be its own month called Adar Sheini (under the single name, Adar but actually more like Shvat) or is it more like Adar itself? Knowing that one may not fast or deliver a hesped during Adar, the Rosh on the first perek of Meseches Megilla (S’7), poses the following question, may one deliever a hesped or fast during Adar Rishon?
(Shulchan Aruch Siman 568 S’7) Logistically, to most, one would think that there isn’t actually any difference. The Rama nonetheless paskins that the first Adar is thought of as part of Adar itself not Shvat, therefore even in Adar Rishon, on the day of a Yartzeit, one may not say eulogies or fast.
The GRa”Ch however, brings a support to the other side in the name of the Mahari Mintz. The Mahari Mintz says, that a boy who celebrates his Bar Mitzvah in Adar (during a leap year) should observe it during Adar Sheini! According to this opinion the month of Adar Rishon is more like Shvat then Adar?! How can we explain this contradiction?
In order to solve our problem, we must say, fasting on the day of a Yartzeit is dependent on the day of the actual death. It just happens to be that during this year there are two Adars. Therefore, hypothetically, one can decide that he wants to commemorate their loss during whichever month he pleases!
By a Bar however the circumstances are a little bit different. In order to be considered a Bar Mitzvah, the young male must be alive for thirteen complete years. In Adar Rishon this boy has not yet lived thirteen complete years, he therefore must wait until Adar Sheini to celebrate becoming a man.
In addition one can say, that one does not truly become a Gadol until he has lived for thirteen years and one day [(Nidda 45b) & (Rambam Hilchos Ishus 2:10)]. He officially does not become a Bar Mitzvah until the first day of the fourteenth year! We can therefore say that in this case, although the boy may turn thirteen in Adar Rishon he does not become a Bar mitzvah until the first day of the next year. He is therefore only considered to be a Bar mitzvah after Adar Sheini.
We can therefore conclude that instead of there being a contradiction in the status of Adar Rishon , practices that are supposed to be observed during this month need to be fully analyzed before being put into practice.*
*As Seen in the Sefer Harrirei Kedem (Thank you Joel for letting me borrow it!)