What is the difference between dancehall and reggae music?

Both are modern Jamaican musics.   Dancehall is relatively recent sub-genre of reggae; the term has probably been around as long the term 'hip hop'.

When you talk about non-dancehall reggae, you are usually referring to what is called 'roots reggae' (such artists as Bob Marley, Culture, Mighty Diamonds).  There is also a variety of other reggae styles like lovers rock.   The characteristic rhythm has the 'skank', which is the sound of the guitar playing the upbeat.

Most dancehall is electronic and turn-table based like hip hop.  The chatting/rapping can be rude and sexual (called slack) but that doesn't have to be the case.  There are plenty of dancehall songs reflecting the spiritual and social consciousness.  Also most dancehall rhythms are more 'dance-able' and can range from sounding very hip-hoppy to soca-ish.  Reggaeton is a genre whose characteristic sound is a branch of dancehall.