No, they don't. If your dictionary is telling you they're the same, it's time to get a better dictionary. Here's what they mean:
a fellow = a man
a buddy = a friend
Who's that fellow over there?
The one with brown coat? I don't know. I've never seen him before.
Who do you usually play golf with?
I play with Joe. He's a buddy of mine from work.
As you can see, they aren't interchangeable - just as 'man' and 'friend' aren't interchangeable. They mean different things.
'Buddy' is a mainly American term. In British and Australian English, we'd be more likely to say 'He's a mate of mine from work'. You may also come across American English men using it as a term of address, as a friendly way of speaking to a man or boy whose name they don't know.
'Fellow' is a little old-fashioned, but it is still used. It's used more in British English than in American English. A more modern equivalent is 'guy', as in "Who's that guy over there?'.
[There is also an adjective 'fellow' which you can put before certain nouns to show that they belong to the same group of people as you. For example, if you are an airline passenger, you could refer to the other people on the same flight as 'fellow passengers'. You could also refer to all the other people who contribute to this site as 'fellow italki members'.]
I hope that's clearer now.