"Recently" means not so long ago (a single event which has ended). "Lately" means beginning not so long ago (an ongoing event which is still happening). Here are some examples.
1) Bob has been jogging lately. (Bob started jogging, and still jogs.)
2) Michelle has been sick lately. (Michelle fell sick, and still is sick.)
3) Peter has been seeing her lately. (Peter started dating not long ago, and still is dating.)
1) Mike finished his studies recently. (Graduating is a single event in the past.)
2) Jamie bought a house recently. (The transaction is a single event in the past.)
3) A new guy moved into my apartment recently. (Moving is a single event in the past)