Recently vs. Lately

I think the difference is that lately has more of a continuous or repetitive meaning (similar to the perfect aspect/tense), and doesn't work well for individual events, while recently can be used for both.

For example,

"I went to the library lately" is ungrammatical, but "I've been going to the library a lot lately" is fine. ("I went to the library recently" and "I've been going to the library a lot recently" are also both fine.)
"I haven't seen her lately" and "I haven't seen her recently" are both fine.
"Have you been to the movies lately?" and `Have you been to the movies recently?" are both fine.

I hope this will help you understand the difference!