Here are sample sentences from this lesson, the first two using ato ni, and the last using ato de:
パーティーの あとに、トム・クルーズの あたらしい えいがを みたいです。 I would like to watch Tom Cruse's new movie after the party.
がっこうの あとに はじめる。 I will know after today's afternoon class.
きょうの ごごの じゅぎょうの あとで わかるよ。 I will start it after school.
I thought maybe the difference is that if you're just saying generally "after" something, you use "ato de", but if you are going to do an action at a specific time "after" something, then you use "ato ni". If this is the case, it seems to be a pretty fine distinction judging by these sample sentences.
This is a good question. I am sorry it took so long to answer.
As for "ATONI" VS "ATODE" it really doesn't matter which particle comes after ATO. What matters most is what you are saying.
If you are saying, "LATER THIS WILL HAPPEN" then ATODE is used BEFORE the action to mean: LATER You will notice with the examples below that ATODE is used at the beginning of the sentence and that nothing comes before ATODE.
EXAMPLES: 1. Atode ikimasu. I will go later.
2. Atode tomodachi ga kimasu. My friend is coming later.
If you are saying "AFTER THIS ACTION THIS WILL HAPPEN" then ATODE and ATONI can both be used to mean: AFTER
- Notice how ATONI / ATODE is used after a verb phrase in the following sentences. - Notice that the verb phrase before ATONI / ATODE is always past tense since the action has to have already occured.
EXAMPLES: 1. Eiga ni itta (atoni / atode) shokuji o shimasu. After we have gone to the movies we will eat.
2. Gakkou ga owatta (atoni / atode) shukudai o shimasu. I will do homework after school has ended.
NOTE: You can use both ATONI and ATODE with verbs like this and the meaning doesn't change. A quick search in google (a reflection of actual usage) shows us which one is more commonly used. ATONI wins over ATODE by an almost 2 to 1 ratio.