In Thanos: Infinity Gauntlet, Thanos was extremely powerful with the Infinity Gauntlet, and would no way get severely injured by Thor's stormbreaker. With the IG, he knew what moves people would make before they made them, he was able to turn characters to glass and shatter them (Thor), turn people into essentially Legos (Nova), Shatter Vibranium with his fist (Captain America), turn Quasar's bands against him, and easily create another person who was nearly equal to him in physical strength. Also, the IG was not even remotely damaged when he snapped. Why was the MCU Thanos nearly defeated?
I'm going to try to flesh this out into a full answer. Lots of speculation ahead.
As I stated in the comments, "Thanos Wrecks the MCU" makes for a boring, repetitive movie. No one is going to want to see that (okay, maybe some people, but not enough to get green-lit by the likes of Disney). A good movie needs to get audiences emotionally invested, but in order to root for an underdog, they need to have something going for them. The MCU accomplishes this by bringing Thanos close to the edge of defeat, showing us that it is possible for the combined efforts of the Avengers, the Guardians of the Galaxy, and others to defeat him.
Thanos in the comics is almost a different character. His trail of destruction up to and including The Snap is to win the attention and affection of Mistress Death. It's all rather...petty, really, but there are different constraints in comics. You can take a long time to get around to your story arc. You can drag out a fight for longer before your audience loses interest.
There is no Mistress Death equivalent in the MCU. Thanos' underlying motives have been established differently - he sees himself as a savior, even if no one else does. He considers himself the only one with the will to do what needs to be done in order to prevent what he sees as the collapse of civilization. This is important, because in the comics Thanos loses because he doesn't believe he's worthy of power. MCU Thanos has no such qualms that we know of - in fact it's almost the exact opposite.
The Snap and the Gauntlet
In the comics, Thanos retains the functional Gauntlet after The Snap. In the movie, the Gauntlet is damaged, which probably serves two purposes. The first is to make Thanos non-omnipotent again (he's already a formidable opponent in combat, with a still-functioning Gauntlet it would become "Thanos Wills the MCU Out of Existence" pretty quickly).
I'm speculating that it also removes the possibility of an "easy fix" for the heroes. A damaged Gauntlet maybe can't undo The Snap with Another Snap, so now they have to figure out another solution. It's possible that the damage will be revealed as something along the lines of "the infinity stones can't be used together again" - we have no real way of knowing. But I believe the damage to the Gauntlet serves as a visual indicator that what Thanos did was a one-time thing.