Late means after the expected or usual time, as in the train was twenty minutes late, while lately means recently, over the period up to now, as in have you seen him lately? If you are talking about timeliness, use late; if you mean "in recent times", use lately.
The core difference
Late describes something happening after it should have, while lately describes a recent stretch of time.
- late — adjective/adverb, after the expected time: He's always late for meetings.
- lately — adverb, recently: Business has been slow lately.
- late — adjective, near the end of a period: They arrived in the late afternoon.
How to tell them apart
Late is the more flexible word: it can describe someone arriving after the expected time, a stage near the end of a period (the late afternoon), or, placed before a name, someone who has died. Lately, on the other hand, only means recently, and it usually sits at the end of a sentence, as in business has been slow lately.
A handy memory hook is that lately pairs naturally with have you, as in have you seen him lately, whereas late pairs with was or arrived, as in the train was late. Mixing them up, such as saying I've been late tired instead of lately tired, is a common error to watch for in learner writing.