To match something with something = to find similarities between two things.
To check something against something = to inspect the accuracy between two things or to examine that which one out of two is more accurate.
I am not sure if I got all the answers right, so I’d better match mine with yours.
This sentence emphasises that you are looking for similarities between two answers to find out if your answer matches with the answer of that person.
I am not sure if I got all the answers right, so I’d better check mine against yours.
This sentence emphasises that you are examining the accuracy between your answer and that person’s answer in order to find out that which one is correct.