Body suits serve several purposes in ponyplay:
In many cases a combination of factors, rather than any single reason, influence a pony's decision to wear a body suit. For me, all but the last reason apply (and the only reason it doesn't apply is because I only own a simple latex catsuit, not a bondage suit). For public events, I think a full body suit provides the ideal way to comply with public nudity laws while still allowing a streamlined, natural look that is not easily achieved by simply wearing tack over clothes (though the latter is perfectly acceptable and certainly much more convenient).
Latex body suits are fairly popular (obviously, especially so with rubber ponies) and look and feel really nice. If I have to be covered up, I will usually choose a latex catsuit (actually I still like to wear one even when I don't have to be covered up). However, it is really easy to get too hot in a latex suit when moving around, and then find yourself shivering as soon as you stop moving (the latex suits are thin). Moreover, latex catsuits can be surprisingly delicate and may be prone to rip, so do be careful when buckling tack.
Of course, if you are allergic to latex, then a latex catsuit is probably not the way to go. Spandex or vinyl body suits are other viable options for ponyplay. You could even go with a thin wetsuit if you like neoprene. It really depends on your personal taste, both the look of the suit as well as the feel of it. An important consideration should be your reason for wanting a bodysuit: are you wearing it because you especially like the material (e.g. latex, vinyl, spandex, neoprene, etc)? Or are you using it solely to cover up for public play?
You may also wish to consider how much effort you want to put into aftercare. Also, some materials will more readily show sweat than others, and possibly more importantly, sweat can permeate some materials more easily than others. Regardless of the material you choose, it is important that the suit fits well and feels comfortable especially if you will be wearing it for extended training sessions (or if you're using it for public play, where it may not be convenient or even possible to change out of the suit for a while).
While most ponies choose bodysuits as a tasteful method to cover up various body parts (or for fetishistic reasons relating to the material the suit is made of), some ponies use them as a form of restraint. There are bondage type suits that force a pony onto all fours by pinning the lower arms to the upper arms and the lower legs to the upper legs within the suit. Generally these are marketed towards puppy play where enforcing four legged status is the norm (whereas in pony play we have many two legged ponies).
However, trainers of four legged ponies might find such bondage suits useful since additional limb restraint is not required when the pony is wearing the suit. Unfortunately, I do not own such a suit nor have I ever worn one, thus, I cannot give any recommendations. I do know of a couple places that sell them, but if you're thinking of spending the $700 - $1000 USD for one of these, I would suggest perhaps soliciting some opinions in the Fetlife forums from current owners. As for places that sell these, I know that rubaddiction sells such a suit as does the shareweb/hot4hogtie store. Both of these are marketed as rubber dog suits for puppy play, but from the photos, it looks like they could just as easily be used for pony play (i.e. they don't seem to have built in dog face masks, paws, etc).
I'm a fairly modest pony, so I will usually opt for a catsuit of some sort even for non-public play. The latex suit that I own (and that is shown in the photos on this page) is a men's latex catsuit from syren. I stumbled across it accidentally one day in the retail location of Stormy Leather. I was reluctant to buy it at first, but it was on sale and fit me perfectly (and covers everything except head, hands, and feet), so I caved.
In retrospect, it was probably one of the best fetish investments I have ever made. I use it in nearly every ponyplay scene (and many non-pony BDSM scenes). Maybe I uncovered a latent latex fetish, but I absolutely love it. It even has "easy access" zippers in appropriate places so, among other things, I can use a butt plug tail with it. I paid around $225 for mine, but it was a clearance sale. This particular catsuit is around $325 normally.
With a little bit of silicone lube, the latex catsuit slides on surprisingly easily (another alternative is talcum powder, but I find the silicone lube to be more effective personally). You will probably need to polish the suit to make it give off that "wet" look, so consider getting some latex polish in addition to the lube. Also, I wash my catsuit right after use because I sweat when wearing it even during short training sessions. I wash it in the shower and use baby shampoo to wash both the inside and outside of the suit, then I simply stick it on a hanger to dry (just be careful when hanging it so that the suit doesn't stick to itself).
Aside from the feel of the suit, I absolutely love that my leather tack does not get sweat on it no matter how long the training session! This is by far my most favored feature of wearing the suit and is why I will brave wearing it in hot weather. It is much easier for me to stomach replacing the less expensive (and not custom made) latex catsuit than to replace any of my harnesses. Nevertheless, I always wash it after use and care for it well. So far it's still going strong.
Anyway, I know this is probably a stupid reason to like a catsuit, but it makes my leather care so much easier. Moreover, I really do love the feel of the latex against my skin, and I do like the way my tack looks against the black latex. Just be sure to wash your latex catsuit after use because in addition to sweat, residual oils from curing and dying leather can permanently stain latex.