The main differences are the nutritional value and amount of calories/sugars.
Alfalfa is rich in protein and is imbalanced in the amount of calcium it carries, so it shouldn't ever be the sole food for any horse (it's actually a legume). It's high in calories, so is not suitable for overweight horses or "easy keepers' as obesity can be a factor in laminitis. And because it's calorie dense and shouldn't be fed in large quantities, this could leave a horse bored from lack of "chew time" if he's stalled.
Fescue, with the fungus or not, is not a great choice for horses. It's super high in sugar (it even affects cows with "fescue foot" ) Its a hardy grass, tolerant to traffic and overgrazing, and it's fattening. However, the sugar content is another reason to avoid feeding it if you can, or at least feed with other less rich hays.
Brome is pretty good. It's one of the higher sugar hays, but is better than fescue.
Orchard grass is also one of the higher sugar hays.
Timothy is about the best "single" hay, in that it's higher in fiber, has a bit less sugar. Still, it's best to mix and match your forages, regardless and not feed a single grass, as that would be as imbalanced as you only eating cheerios day in and out.
Bermuda hay is great. It's low sugar, so perfect for lamintic horses.
That's a sampling. And of course, the amount of starchy sugars can vary within a type of hay, due to weather, time of day, etc. The best thing is to have your hay sampled to see what is in each batch, and you can supplement whatever nutrients are lacking. Remember, variety is important. A small amount of any of the "richer" hays is fine, if balanced with the more filling," lower octane", and may eliminate the need for supplements. If your horse isn't getting enough calories from hay, you can try feeding the richer hays before resorting to grain.
There are also forage alternatives, but that's another ball of wax.