Hi Dana!
Like Michael said, it's a subtle difference. They're very similar.
When you say "not especially", it means the subject is likely true, but not *very* true. It is more about a *degree* of something. An example:
"It's not especially hot today" meaning: it's hot today, but *not very hot* (and may be a little cool).
If you say "It's not exactly hot today", the meaning is more: it's definitely not hot (and probably is cool).
They're very similar, and you can usually use them interchangeably. It's just a subtle difference.
Hope that helps!