To add a few more examples, they are not often interchangeable. You would tell your mother to "check" to mail to get letters, but you wouldn't tell her to "examine" it, because then she would just be standing outside the mailbox, staring at it. You would, however, ask the doctor to "examine" you if you are sick with something, because it doesn't make sense grammatically to ask your doctor to "check" you.

They are often different. Checking something is often done less carefully than examining that same thing. When you examine something, you are look at it for a very long time, whereas checking could be just a quick read to spot mistakes. Just a few examples of the usage of these two words: "The teacher checked the student's exam paper." -- The teacher is just looking at the exam paper, probably to look for mistakes/corrections. He is probably not going to be checking the paper for a long time. "The teacher examined the student's exam paper." -- The teacher is really, really looking at the exam paper, scrutinising it for a loooong period of time. Perhaps there is something suspicious about the exam paper, eg. the student cheated and the teacher needs to verify to see if it is true.

in what context?