When you tolerate something, you let it exist or happen. You allow it. Generally, it means that you neither approve nor disapprove.
Ex: Jason's new drum kit made more noise, even in the basement, than his mother had expected. She tolerated it for three weeks before she told him he'd have to move his drums out to the shed.
Ex: Our company has started to dress down on Fridays. On that day only, jeans and trainers are tolerated in the office.
You will often encounter the noun form of the verb tolerate in the form tolerance, which conveys the ability or will to let something happen. It is a synonym for acceptance and is often applied to the way people react to differences. In that sense, tolerance means having a fair, accepting, and objective outlook towards others.
Ex: The school has adopted a zero tolerance policy on cheating. Anyone caught plagiarizing automatically fails the project and, of course, may be expelled.
Put Up With
Putting up with something means that you are accepting or continuing to accept an unpleasant situation. You would not choose the situation, but it is not so unpleasant that you need to take action.
Ex: The twins' mother told them that she wouldn't put up with any more fooling around while she was driving, because if she got distracted it could cause an accident.
Ex: I had to put up with my husband's snoring for hours last night. He's got a cold, and when he has a blocked up nose, he always snores.
Put up can refer a number of things, including making a display of something. And you might say put up or shut up when you want someone to either do something about a situation or stop complaining about it.
Ex: John was worried about how he'd done on the algebra test, but he managed to put up a brave front for his parents. In front of them, he stayed calm and acted as if he knew he'd done well.
Ex: The rugby coach was upset with the amount of effort his team had shown in the first half of the match, although they'd all talked about how this would be an easy victory. He told them that it was time to put up or shut up.