There are no one-to-one translations here. It really depends on who is talking to who and the context of the conversation.
after all in these two sentences are similar, but they take slightly different meaning.
The first sentence implies that they were aware of some indications or expectations of snowing. Maybe they had a chat about whether it is going to snow. It is close to saying
You should wear a jacket. After all, it actually snowed (as we expected or didn't expect).
In the second sentence, the speaker uses
after all to emphasize the fact that it is their anniversary today.
I would translate these two
after alls differently in Japanese. Here is my take.
You should wear a jacket. After all, it's snowing out there.
だって would also work depending on the context, but
やっぱり sounds more natural to me as it describes a situation that something happened contrary to their expectations.
Of course I bought you a present! It's our anniversary after all.
Using ～じゃないか emphasizes the fact that it is our anniversary. These below may also work. And you can also choose to prefix each of these with なんてったって.
Sorry for a very personal answer, but I think it really depends on the context (and also who translates it).