It is my understanding that both these sentences have very similar translations but I don't quite understand the nuances:
おわった あとで 行きます。
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てから sort of imply a more strict chain of events
I will finish then go.
あと is more vague and only imply that event A happened after event B
I will go after I finish.
In most case the meaning will be very similar, but あと will have more leeway. It could be used to imply something will be done after, but maybe not immediately after. It could also be used to stress the order of event and make it clear what happens before and what happens after.