Both can suggest the idea of a long wait or delay e.g. When at last they found him he was half dead. Or After 3 attempts Alice finally passed her driving exam. (eventually is used in same way) Only finally can be used to introduce the last idea in a series e.g. When you leave take a key, a bag, a coat and finally a scarf. Both can be used as an exclamation. At last! Where have you been? Finally! What happened? 'In the end' is another expression you can use to show something happens after a lot of uncertainty. Did they get married in the end?

In some contexts they mean the same thing. For example if you have been waiting for a long time in a restaurant for your friends to come, when they arrive you could say: Finally! You're here. At last! You're here. Or you could write something like 'Finally his friends had appeared.' You could also write 'At last his friends had appeared.' Here you use them in the same way. But if you mean 'finally' to mean the last thing then its usage is not interchangeable with 'at last.' For example 'Finally, I would like to say that I am very pleased that you have all come here today...' You can't use 'at last' instead of finally in this context. Un saludo.