"I hope you and your wife can come together."
Is there a difference between 能 and 会 that makes this one have to use 能 to say "can"?
Yes. 能 (neng2) refers to a physical possibility. 我能去派对 = i can go to the party --- I have the option of going. 会 (hui4) refers to a learned ability. 我会说中文 = i can speak chinese --- I have learned how to speak chinese. (My mnemonic is that HUI sounds a bit like HOW: 我会做... = I know HOW to do...
I have some personal preference but I think they are not rules. I would say 和 and 跟 are pretty interchangeable.
have no difference.
However there is the word Can in this sentence. There is a grammatical difference between 和 and 跟 as 和 is a conjunction while 跟 works like an auxiliary verb (the meaning is "follow", or more conceptually "team up with", "group up with"). This would require different grammatical syntaxes in the sentence, which are:
(我希望你跟你妻子能一起来 is also viable but focusing on whether the listener can come together with his wife or they have to come separately.)
It would seem pedantic to some people, but I think these are the correct ways to say if anyone would like to sound perfectly natural.
Not a native, but wouldn't there be a 我? Also, would 我希望你跟妻子能一起来 be correct?