Well, that's even hard to explain without being an expert in sciences dealing with these creatures, but let me try.
'Kummitus' quite specifically refers to ghosts residing in haunted places. There is also a corresponding verb 'kummitama' ('majas kummitab'='the house is haunted'). I don't know the etymology of this word, it could somehow be related to 'kummaline' ('strange, peculiar').
(Most certainly not to 'kumm' ('rubber'), though with a vivid fantasy one could find a connection there, too).
'Vaim' could also be translated as 'spirit'. In this context it mostly describes ghosts/spirits of the deceased (e.g. 'Hamleti isa vaim', 'the Ghost of Hamlet's Father'), but sometimes also a ghost more generally. There is some connection with 'kummitus', e.g. I think it's ok to say 'vaim kummitab', especially if you specify whose 'vaim' it is. It also works as 'spirit' in other contexts: 'koostöövaim' ('the spirit of cooperation'), 'samas vaimus' ('in the same spirit/manner'), etc.
'Tont' is kinda hard to specify. I would use it as a general word for 'ghost' (in that sense it partially overlaps with 'vaim'), but not in the more specific meanings described above. It sounds a slightly outdated and fairy-tale word, but perhaps not entirely so.