They are almost the same and can be interchanged in most cases, however there are specific cases you would use recently instead of lately, that applies to a specific event/action. For example: correct: I got married recently. wrong: I got married lately. The second sentence is incorrect because the action/event of "getting married" is not something you do on a regular basis.

It's fine to say 'I went to the library recently'. This refers to a single visit in the past. In this case, 'recently' means something like 'a few days ago', so you can use the past simple form here with 'recently.' You can't say 'I went to the library lately'. This is an incorrect sentence. 'Lately' refers to a period of time (maybe the last few days. the last few weeks) so you have to use the present perfect. For example, you can say: 'I've been to the library a lot lately.' or I haven't been to the library lately. This second sentence means the same as 'I haven't been to the library recently'.