Am I missing something with Winston's Tesla Cannon?
Solution 1:
His damage is one of the lowest (if not the lowest) in the game. Here is a relevant quote mentioned by Blizzard developers (This is in reference to D. Va, but the Winston mention is specific).
D.Va's damage is definitely on the lower side, much like Winston's. They are this way for a similar reason: They are both very mobile and hard to kill. Every character in the game has strengths and weaknesses, its part of what makes the teamplay work well.
Source: http://us.battle.net/forums/en/overwatch/topic/20744914371?page=3#post-43
As for "missing" something about the mechanics of Winston's cannon, there are a couple of things to keep in mind.
It's an AoE. His cannon can hit multiples targets in front of him. It's a fairly small cone, but it will hit more than one enemy at a time if they are grouped up.
There is no need to aim (mostly). His cannon doesn't really need to be aimed, just pointed in the enemy's direction. It can hit blinking tracers and such with nothing required except making sure you are looking at them and within range.
His jump also does some damage. Minor, but if you jump ON your target, they take damage, then you burn them with the cannon and can jump away if it looks like you're losing, which ties in to the developer reasoning above.
Explicitly to answer the question, Winston's cannon will do more damage by hitting more than one enemy at once. Winston isn't bad overall, but he is bad at dealing damage - that is NOT what he is designed to do.
Solution 2:
The advantage of the Tesla Cannon is its wide cone of fire, which makes it able to easily hit agile targets like Tracer and mechless D.Va, as well as hitting multiple enemies if they're all in the cone. This is particularly handy when entering a Symmetra turret nest, as a quick sweep will destroy all of them immediately.
The DPS is definitely on the low side, but it's usually enough against frail characters that Winston loves to jump on, like Widowmaker or Mercy. Winston's strength comes from his high health, shield to isolate himself from the majority of the enemy team while behind their lines, and booster jump to drop on a target and keep chasing while dealing constant damage.
Solution 3:
Winston excels in mobility and getting out of a bad fight easily. He also excels in what I like to call "personal combat arena".
What I mean is, through the use of his easy mobility to get to the other team's backlines, especially where Widows will usually lurk, he can hop in, plop a shield down, creating a mini space where he can fight the enemy without fear of getting chipped from other team members of the enemy. His mobility also lets him do a lot of work going against Mercys who are flying around to other team members, making taking down the medic a lot easier.
That is why Widowmaker's 2nd counter, besides another Widow, is a winston, because although his weapon is a very slow DoT, Winston has a huge HP pool, can shield up to create a personal area to fight in, and can easily jump back out should his HP get too low, then jump back whenever ready.