The sentences are not the same and 'they are less/more likely to stay' are not complete expressions. They will 'probably' stay means that it is quite likely they will stay (e.g. approximately 60-70% chance. 'More likely' or 'less likely' should be used to compare one thing with another which is why I say that it is incomplete. I would use it as follows: 'They are more likely to stay at our house if we have a bed free for them' (i.e. comparing the probability of them staying with and without a bed being free)

I would say they are pretty much the same in terms of everyday English. But we don't generally say "they won't probably stay " we say, "they probably won't stay".