They can both be used to intensify, but they're not always interchangeable. I can say:
"I really like JLG's answer."
but I wouldn't say,
"I very like JLG's answer."
Correspondingly, I could say,
"I like JLG's answer very much."
but I probably wouldn't say,
"I like JLG's answer really much."
(That's just an example, even though I really do think it's a very good answer.)
That parenthetical statement brings up one other distinction: really can mean truly, as well as very. So, if I say,
"I'm really proud of my daughters."
that could mean two different things. It might mean, quite simply:
"I'm very proud of my daughters."
or, it might mean,
"No, I'm serious – I truly am proud of my daughters."
(The latter might be used in the case where, say, I had told someone that I was proud of my daughters, and they seemed somewhat incredulous, unaware that I even had a family.)