John is a hunter. He loves to hunt for small animals.
Jack is the owner of a forest, "Green Forest", where he allows people to hunt for a pay.
Both John and Jack wish for there to be more small animals in the Green Forest. It's in the John's interest because he loves hunting. It's in the Jack's interest because the more animals there are, the more profit he can generate from the forest.
So they have one common interest.
But they both also like painting. Now they have two common interests. They meet to discuss different artists. They don't benefit from this directly, at least not in cash terms, but still it's their common interest.
Now, according to the Cambridge Dictionary,
to have something in common - to share the same interests or have similar characteristics. ("I didn’t think Larry and Patricia had anything in common, but they talked all evening.")
Probably this latter idiom is "wider". It covers not only similar interests, but similar personal traits.
Here's a quote from a book:
Stalin and Hitler had many things in common. Both had alcoholic fathers, were radicals at a young age, and were very ambitious.
Hitler and Stalin had many things in common, including some common interests, like helping each other divide Poland. They get on well until 22 June 1941, when Hitler attacked USSR.
After 22 June 1941, they still had some common interests. They both liked American movies, for example. But they lost some other common interests, and because of that they did not get on well any longer. They still had many things in common, though: in addition to their remaining common interests (loving US movies), they still were very ambitious, they still were sons of alcoholic fathers.
Let's imagine they lost all common interests on June 22, 1941 - including their love for American movies and stuff like that. We would still say that "they have many things in common". We would've said: "They resemble each other in many regards, but they have no common interests whatsoever".