Any difference between:
"He is an important customer of ours"
and:
"He is our important customer"
Any difference between:
"He is an important customer of ours"
and:
"He is our important customer"
Yes, it's pretty different.
The first sentence means "He is one of our important customers". You have several important customers, and he's one of them. "ours" refers to "our important customers".
On your second sentence, you're saying you only have one important customer. It's technically correct but I don't think it's employed a lot.
When someone has more than one of something, we can use a/an + noun +
So, if you say "he is an important customer of ours", you mean he is one of your important customers. If you say "he is our important customer", there is nothing in the sentence that would imply that you have other customers. You just state the fact - that customer is important to you.