The only difference is that one is singular and the other is plural. That's all! Use the singular when there is one single interest that is shared. For example, you say "Bill and Jack have a common interest in music" because music is a single interest that is shared by them. The fact that "music" is singular has nothing to do with it. For example, you can also say "Bill and Jack have a common interest in music and dance" even though "music and dance is plural". That's because you are still thinking of just one interest that they share: their single interest in music and dance. However!!! You could also use plural and say "Bill and Jack have common interests in music and dance." Both ways of saying it are correct because you are free to think of their interest(s) in music and dance as one interest or as two interests. Choose the way that feels best to you. You say "Clean streets are a common interest", even though "streets" is plural, because "clean streets" is a single shared interest. If you said "clean streets are common interests", it would sound strange (but not ungrammatical) because it would be odd (but not wrong) to think of each street as a separate interest. Use the plural when there several interests. For example, "Bill and Jack are friends because they share common interests: art, sports, and photography".

’It is in our common interest that we come to an agreement.’ ’They have a common interest in resolving this problem.’ ’I would have thought it was in their common interest to pursue this.’ The ’common interest’ is their mutual benefit; both [if two] or all [if more than two] of those involved will gain something. What these benefits are will be explained in the accompanying announcement. 'Our common interests include improvement to transport infrastructure and investment in publicity and marketing.’ NB ’Our areas of common interest include improvement etc..’ ’They have several common interests to be discussed in the forum.’ NB. 'They have several topics of common interest to be discussed.’ In all these examples ’common’ means ’shared’ or 'mutual’, and ’interest’ means 'of benefit' in ’common interest, and ’issues we/they want to discuss’ in ’common interests’.

'In (someone's) common interest' is often used in the sense of mutually beneficial. e.g. "Brining industrial action to an end as quickly as possible will be in everyone's common interest" = mutually beneficial for everyone. 'Common interests' would be more usual when speaking of multiple things that the parties have an interest in. The countries in the EU have many common interests.

Between companies or countries, it would always be in the plural “common interests”. I’m sure for obvious reasons. There is always more than one interest in common.  in fact, I would say, in most cases it’s used in the plural, even between friends there is usually more than one thing in common.