The main purpose of heavies, early on, is a panic button: if you pull too many sectoids, thin men, and floaters all at once, you can nuke a fair number of them with the heavy's missile, with a near 100% chance to hit (the tooltip says 90%, but most of the 10% misses land very close to the target spot, and usually hit everything you were aiming at). It's also quite effective against mutons hiding behind cars, vans, or trucks, since the secondary vehicle explosion happens immediately.
They're also great when you have tough enemies hiding behind destructible cover, and want to improve others' aim by blowing it up.
Mid and late game, they're your primary means of taking out deadly robotic units via shredder rockets and/or heat ammo.
Often, when you build your first heavy, you'll be tempted to take the wrong skills, because some of the worthless ones sound pretty good...I know I took almost every wrong skill on my first one, so I think a big part of knowing what role a heavy should play is knowing how to build one:
At corporal, Bullet Swarm makes them very effective when you have a swarm of enemies advancing on you, and helps to make up for the fact that heavies have the worst aim of any XCOM soldiers. Holotargeting is usually not as useful: a hit from a heavy is worth more than a hit from any other class (except sniper), more shots generally results in more damage than a measly 10% aim bonus against one unit, and there are better options (Bullet Swarm, Shredder Rocket, the assault's Run and Gun + Rapid Fire, the sniper's Head Shot) when you need to quickly focus something down.
At sergeant, Shredder Rocket gives you a second panic button against large groups of weak enemies, and an excellent way to deal with deadly robotic units: hit it/them with a shredder, and the damage amplification will help the rest of your crew take it/them out quickly. Suppression is generally not as useful, in part because an alien that's dead is less likely to hit you than an alien with -30% aim, in part because it burns too much ammunition: it requires two of your heavy's three ammo to use the skill, meaning you lose the option of using Bullet Swarm on the following turn, and you can't keep an enemy suppressed for more than one turn. It often leaves you out of ammo in deep trouble. On the other hand, Shredder Rocket gives you an additional option when you're out of ammo. The aim penalty on the reaction shot (if it happens) is also pretty big for a heavy, whose aim isn't very good to start with.
Compare that to a support's Rifle Suppression, which a support can use two turns in a row even before Ammo Conservation, which doesn't prevent taking two shots in a turn (since a support can't anyway), and remember the support's aim is second only to the sniper, so the reaction shot penalty isn't as big a deal, which makes the support usually the better option for suppression, if you like to use it.
At Lieutienant, HEAT Ammo is the clear winner over Rapid Reaction, and the number one skill that makes heavies worthwhile: do you want two shots of reaction fire, given a heavy's low chance of connecting on a reaction shot? With the second shot fired only if the first one hits?? To boost the rate at which your heavy runs out of ammo??? Or do you want double damage against the deadliest enemies in the game, and the ability to take one out single-handed if both shots from bullet swarm hit?
At Captain, both choices are an option: grenades are another guaranteed hit area-effect weapon, which is useful given the heavy's poor aim, and, like rockets, they're useful for destroying cover and exploding vehicles that aliens are hiding behind for bonus damage. If you like your heavies to carry grenades as their accessory, it's the obvious choice.
On the other hand, two extra tiles' radius on your rockets from Danger Zone makes it easier to get large numbers of enemies in the blast radius, and is the only option if you prefer to use a scope to boost the heavy's poor aim, or anything other than a grenade.
The bonus radius on suppression would almost make for an argument to consider suppression, but in my experience, aliens rarely end their turn within two tiles of another alien, so suppression remains single-target most of the time. (If you could aim the diameter five area of suppression fire, instead of having to center it on a specific alien, that might be different, as you'd often be able to suppress three instead of just one if you could center the suppression fire on open space.)
At Colonel, a third rocket is generally more useful than +2 damage on the two rockets you already have.