The meaning is the same, but the implications are slightly different. "This door doesn't close properly" implies that this may be a permanent situation. It is a statement of fact about the door. "This door won't close properly" suggests that the door is 'refusing' to close. This could be a temporary situation, for example: 'This door won't close properly'. 'That's because there's a bag on floor just behind it. Look!' 'Oh, yes, of course. I didn't see that bag. I'll move it away.'

They are different because of the verb tense you are using. "WILL not close" indicates the future "Does not close" indicates present

There is little difference in the meanings of the sentences. You can use either in the case of a door not closing properly, although Will Not might imply that there is no obvious reason for the door not closing properly, whilst Does Not would imply that you can see what the problem is.