I think i've figure out the nature of Tom and Jerry's fairly complex relationship. It's close to that of a younger brother and an older brother.
Tom is the older sibling. He's the the one that's constantly trying to stop Jerry from reeking havoc everywhere he goes. Jerry always bothers Tom whenever he's on dates or hanging out with friends, Tom just wants him to go away. Jerry does whatever he wants, not caring about the consequences or the effect it has on Tom. He also gets annoyed when Tom comes to put an end to his fun, even though he can be a bit of a wet blanket at times. But sometimes, on rare occasions, they put they're differences aside and work together to fight against one common problem. And, if all goes well, they enjoy each other's company, even if only for a short while. And than they start fighting again, and the cycle starts over again.