In your example, "abide" and "bear" are interchangeable and mean "to tolerate" or "to put up with" something difficult, but you will only hear "I can't bear it" ("I don't bear it" is correct grammar but we don't say it that way.)
"Abide" can also mean "to continue to exist", especially when bad things are happening. This meaning is more passive than "to tolerate" something, it's like standing in one place while events happen around you. It's considered old-fashioned and isn't commonly used.
"Bear" can also mean to carry or support something heavy, and this is how you will hear it most often. For example: the column bears the full weight of the roof.
In the spoken language, we use the idiom "put up with ___", for example "She doesn't/won't put up with that kind of behavior."