The first sentence is actually just incorrect... Having done his homework, George went out and had a long-awaited beer. You don't need the "after" because the idea of completeness exists in the choice of the tense. You'd use "having done" to connect the completeness of the homework to the action that comes next -- the beer. This is connecting past action to present conditions. For example, Having won some money, I started thinking about which car to buy. For the second sentence, the "doing his homework" is a gerund (a noun phrase made from verbs in different forms)...

In the example you gave they are the same thing. "Doing" can be present tense (I am doing my homework) but it can also mean past tense (finished) if you include "After" before the verb. Some examples: "After doing the dishes... = After having done the dishes..." and "After watching TV = After having watched TV...". Also, in American spoken English it is fairly rare to hear someone say "After having done."

That said, speaking is different...I could imagine someone saying that first sentence and it'd sound okay. However, in comparing the two sentences in writing...that first one is awkward...