Activities and periodic action (particularly in one of the perfect tenses, including perfect progressives) go better with "lately". Punctual events go better with "recently". In fact, in a sort of impressionistic way, "lately" is imperfective and "recently" is perfective. Hence, "lately" usually means "within the most recent period".
Neither goes with the present tense very well.
He coughs lately.
Recently she is studying more than usual.
In the simple past, it's "recently".
He returned from Europe recently. [NOT "lately"]
He recently won an award for poetry. [NOT "lately"]
I recently lived in Dallas. [NOT "lately"]
In the present perfect, it's either one, depending whether an activity or event is the focus, the progressive tense focusing on the activity.
I have recently read "Romeo and Juliet".
I have been reading "Romeo and Juliet" lately. (or "recently")
All in all, the most typical uses are Past + recently and Present Perfect Progressive + lately.
I'm sure there are other patterns that my quick brain sort may have missed, but that's why we have lots of moderators!