Well good on you for wanting to know...!!
Hay is just grass which has been cut, left in the field, turned a couple of times, and then baled up with no wrapping - it can be dusty and full of spores, especially if it's kept a long time, and that can cause respiratory problems, also if it's still a bit wet when baled it can go mouldy.
Haylage (the name comes from silage, from the French 'en silage' which means wrapped up) is sort of halfway between hay and silage in wetness, but these days i think far more of it is dried out like hay, and then wrapped.
Because of the wrapping it will ferment slightly and can't be eaten for about 6 months after wrapping- after that it's ok. As I said a lot of it these days is drier than when it was originally started, it won't have spores or dust so is far better than hay for horses with breathing problems.
The wetter stuff is more energy giving, but you need to feed more of it than hay, weight for weight, which sounds somewhat contradictory, I always think.
I've learnt a lot about haylage as we have part of our field cut for it every year, so did a fair amount of research - we leave ours out 'til it's as dry as hay, and only then do we bale it up and wrap it - well,a local farmer does it actually, and we pay him for doing it, but it's still cheaper than buying in.
That way we can also store it outside, as we have no barn to put it in.
Hope that's helped :o)