word choice - What's the difference between に and で when speaking of time of an action?

What's the difference between に and で when speaking of the time of an action? I know に is very specific about time, but I'm not sure when, or how to use で. Can で only be used in certain instances?

Supposing we want to say "After eating breakfast, I will watch TV." What would be the difference between these two:

朝ご飯を食べた後にテレビを見る。(asagohan o tabeta ato ni terebi o miru)

vs.

朝ご飯を食べた後でテレビを見る。(asagohan o tabeta ato de terebi o miru)

Thanks!

Answer

There are some interesting connotations in the Chiebukuro examples crunchyt kindly pointed to which I think are worth going over in more detail. First, the ～でする and ～にする forms:

この仕事はあとでします。 I'll do this job later.

この仕事はあとにします。 I'll do this job later.

Now, the fact that these are both allowable and both have the same (English) translation doesn't mean they're freely interchangeable. Rather, the first simply means that you'll do the job sometime in the future. The second, however, implies that you have the option of doing the job now, but you're putting it off until later -- "you choose 'later' from amongst other options," as crunchyt put it. (Incidentally, the Japanese word for "postponement" is 後回し. Yet the opposite of "postponement" is not 前回し, but 前倒し. In Japanese, things rotate backward and fall forward. :)

Now for verbs other than する, で is the typical choice to follow あと. From Chiebukuro:

ご飯はあとで食べます。 I'll eat later.

ご飯はあとに食べます。 (incorrect)

But what of the ～たあと form? If you Google for such phrases as 食べたあとで versus 食べたあとに, you end up with the same number (~8M) of results. Clearly both are used commonly. What do we make of this?

Recall that に refers, as you know, to a specific point in time. で, on the other hand, is more general, and indicates a range of time within which an action takes place. For example:

この本を３時間で読み終わった。 I finished reading this book in three hours.

この本を３時間に読み終わった。 (incorrect)

From this contrast of specificity/generality we can conclude that ～たあとに shows a tighter bond between the two actions, i.e. the second action takes place very close to the end of the first. ～たあとで, by giving a range of time, is much less specific about when the second action starts.

This rule of "immediate versus range" also applies to cases where the word preceding あと is not a verb, but one of この, その, or あの:

このあとに [right] after this

このあとで [sometime] after this

There are of course cases when neither ～あとに nor ～あとで may be used, such as when the first action is a prerequisite for the second:

歯を磨いてから寝なさい。 (not 磨いたあとで)

Or when the first action has a direct cause-and-effect relationship with the second:

このボタンを｛押したら／押すと｝お湯が出ます。 (not ボタンを押したあとで)