You should pay him a visit and see what he has to say. And

You should pay him a visit and see what he says.

What's the difference between these two sentences and could they be used to suggest to do something in the future?
Yes, they can be used to suggest doing something in the future.

I'm not sure I want to fall into the "what's the difference" trap again.
In a certain sense, there is no difference.
In terms of common usage, "see what he has to say" is usually understood to mean " . . . . about some specific thing."
Depending on prior context, "see what he says" could mean this too.
Or it could mean just "see what he says."
On the other hand, "see what he has to say" could also mean exactly what it says.
It all depends on context.

If you really want to know the difference, "what he has to say" is more likely to mean "what he has to say about something."
The two sentences have basically the same meaning. They are two ways of saying the same thing.

what he has to say is what he can offer as his opinion (on some topic of interest to you).
what he says is, well, what he says, the words he speaks (on some topic of interest to you), which will be, as in the previous sentence, his opinion.

There is only this slight difference: In the first case, the speaker might think that 'he' already has an opinion "stored up" (he has it to say) that he will share with the speaker. In the second case, the speaker does not necessarily believe that, but may believe that 'he' will come up with something to say spontaneously at the moment he is asked. In spite of this slight difference that we can detect by taking the sentences literally, in fact, both sentences are used in the same situations and are virtually interchangeable.