I have a lot of the Malaysian Trumpet Snails (MTS) in my heavily planted 39 gal tank. They do not damage the plants. They are beneficial in that they will burrow through the upper layers of the substrate, very earthworm-like, but without damaging roots. They are shy, and mostly nocturnal. After lights out on my tanks, I can see them rise from the substrate, up the side of the glass (quite a troupe of them now!), and start tooling around for the night.
I think they're great, and have introduced them to my other planted tanks.
There are two main varieites. Both have long, conical, trumpet-shaped shells, and average 3/4" long, not exceeding 1.25" long (the 2 yr old "big mama" of my tank). They are trapdoor snails, meaning they have an operculum which they can shut tightly. They are also livebearers, meaning they give rise to fully formed, but miniature baby snails, rather than laying eggs. The plain type has a dark grey to brownish shell, with little ridges around the opening. The "mocha & cream" type have a light tan shell, with darker brown flecks throughout. The ridges at the opening seem less prominent to me. They will interbreed, too, producing a snail with a medium dark shell and slightly darker flecks.
HOWEVER, be aware that once you decide to introduce MTS, they will be with you until you tear the tank down.
Because they burrow, they would be hard to eliminate from the tank, as there would almost always remain a few that are buried and can elude you. As they are trapdoor snails, they can survive loaches trying to eat them. Being livebearers, they reproduce quickly, and are not subject to egg-eating fish. Another caveat is that if solitary, they can self-fertilize.
So, consider carefully if you really want them.
That said, I am a big fan of MTS. They do not damage my live plants, yet will eat debris, including dead stems of plants that die with normal growth turnover. I also have the mini ramshorns, which are egglayers. I have few of them, as the sparkling gouramis eat the eggs.
Other well regarded algae eating snails are Olive Nerites. These will lay infertile eggs. They only successfully reproduce in brackish water, so you're guaranteed to be able to control the population.
Apple Snails eat plants.
Ditto for "Mystery" Snails (brigsii). Brigs will prefer other food, but aren't above making a salad of your plant centerpiece.
Columbian Ramshorn Snails (the big 1"+ ones) will DESSIMATE plants.
Pond Snails will eat algae until they run out, then will eat plants. These guys have a football-shaped shell. They will also reproduce like crazy. They do not have an operculum though, (can't close a trapdoor) so loaches will happily feast on them. Early on, I used a loach brigade to get rid of them. I think the MTS are highly preferable. (Go Loaches!)
In a heavily planted tank, algae eating shrimp are handy, too. I keep a fleet of cherry reds. They're always busy delicately picking at the plants, and my tank is remarkably algae-free.
Hope this helps!
-Jane