Slow describes a thing or person (a slow reader, a slow car), while slowly describes how an action is carried out (she walked slowly). If you can ask "how?" about the word, you likely need slowly; if it is describing a noun, use slow.
The core difference
Slow is chiefly an adjective (and a verb), while slowly is the adverb that tells you how something happens.
- slow — adjective, not fast: The traffic was painfully slow this morning.
- slow — verb, to make something happen at reduced speed: Please slow the car down.
- slowly — adverb, at a slow pace: She walked slowly along the beach.
How to tell them apart
Ask what the word is attached to. If it describes a noun, such as a slow reader or a slow train, you want the adjective slow. If it is telling you how a verb happens, such as walking, working, or improving, you want the adverb slowly, as in the medicine works slowly.
The mix-up shows up most in casual speech, where slow is sometimes used informally as an adverb, as in drive slow. In careful writing, keep slow for describing things and slowly for describing actions.