"Eventually" just means that something occurred after a [long] time, whereas "ultimately" refers to something that occurred at the very end. So, if you're telling the story about a man who wanted to go to the supermarket to buy milk but was too lazy to get up, you could say: "Despite his laziness, he EVENTUALLY got up out of bed, but ULTIMATELY did not end up going to the supermarket." So, "eventually" explains that after a long time, he did indeed get up, and "ultimately" explains that despite that, in the end, he did not actually go out to buy milk. I hope this helps/makes sense. :)