Learn English – Usage of “anyone” and “anyone else”

There are a few contexts, such as...

1: Does anyone [else] want a cup of tea?
2: I'd rather die myself than kill anyone [else]

...where including else radically changes the implication. In the first it's certain at least somebody (probably the speaker) wants a cup of tea. The second would only be likely to be said by someone who had already killed at least once before.

But usually (as in OP's context) there's no real semantic effect (the meaning doesn't change). It's just that else adds a touch of emphasis (even more so if it's better than anyone else in the whole wide world).

EDIT: I've just realised that my example re touch of emphasis above is potentially misleading. Including else doesn't actually imply anything like ...in the entire universe. What it does is draw attention to the fact that no alternatives are better - which often implies contextually-relevant alternatives. Thus one workman might say to another Why don't you ask the boss for a raise?...

3: ... You know him better than anybody
4: ... You know him better than anybody else

It's a fine point, but I'd say the first version (which might even feature exaggerated stress on anybody) can more easily carry ...in the world as an intended (but unspoken) meaning, where the second more naturally implies ...in our workgroup. That's to say, we the workmen are the contextually relevant group within which no-one else knows the boss better than the addressee (and it's quite possible none of us really know the boss very well, but somebody must know him at least a little better than anyone else in that group).