No, we did not evolve from monkeys. Humans and monkeys evolved side by side from a common ancestor that looked a lot like a shrew that was a tree-dweller. They both branched off from these ancient tiny primates and were never on the same branch together. We are part of the ape family. Apes have no tails, they can swing through branches, unlike monkeys, and I think there is a difference in their noses. I think apes have a difference in the skeletal structure of their noses but I don't remember positively. The body structure of monkeys is a lot more cat/dog-like than apes.
Our brains started to grow, leading us down the path of dominance because we discovered fire and began eating cooked meat. It's easier to digest and left more energy to put into growing our brains.
Other apes have fur, not just the monkeys. Fur protects from the crap that comes out of trees (I assume,) pieces of bark, and stuff like that. Once we were on the ground, we didn't need that protection. The gene is still with us, but is inactive. We know this because of people with hypertrichosis. When fur is not needed anymore, mammals lose it. Look at cetaceans, the whales, dolphins and porpoises. They lost most of their fur. They just have a few scattered whiskers left around their mouths. Naked mole rats lost their fur probably because they live underground and being furless makes it easier to burrow and run through the burrows. I just read that another reason humans could have lost their fur is to be rid of fur-infesting parasites.