'recently' has two meanings:
1. with "I saw" (simple past) it means "a short time ago".
2. with "I have seen" (present perfect) it means the same as "lately".
'lately' is primarily used with the present perfect, whereas 'recently' is use with both the simple past and the present perfect. Given the choice in,
EX: I haven't seen him _______ . (present perfect)
I'd use 'lately'. It goes with have seen.