It's not that you absolutely need snipers, with enough smoke and ammo spraying you can get through the game, but there are a great many maps where they are useful.
On almost any map that got open places at all. If there are civilians close enough to aliens to be in danger, we need to remove the threat on the same turn, if at all possible, thus at long range. And if it's too far for smoke grenades, the only option is plain killing.
Rocket launcher got very long range, but not very precise and deals splash damage.
Of course, a long burst from MG is fine too, but it takes more TU - a sniper have time to move a little, crouch and pop the target, while MG is limited to short burst after move, so at long range it will be just 1-2 hits tops, not lethal enough. Even worse vs. armor. And again, when you need to kill at long range, you need accuracy to avoid hitting civilians. So...
I take 2 snipers early on. Sniper rifle deals much more damage than AR, one shot usually kills unarmored aliens.
Laser rifle got betteer range and precision, and even if it's weaker than sniper weapons, still more damage than AR - but it needs to be developed first.
You may also want to check your soldier stats, a sniper with 22 accuracy and 21 weapon skill is a total putz who can be out-shot by a grunt with a shotgun. Your good snipers will start with a 27+ accuracy and a 26+ skill (I also prefer a 35+ strength)
Strength matters only when there are EM rifles. With starting sniper rifle it means ability to carry a medikit and an extra smoke, not that it wasn't a nice bonus.
Now, whether a sniper should wear armor... yes: it allows to use them on medium range where reaction fire from a plasma pistol still gets them occasionally.