Can I say "What's the sense of (something)"?

I came across this usage in a sentence from an exercise, which is:

What's the sense of having a public open space where you can't eat, drink or even simply hang out for a while?

Does the usage here sound natural? And how is it different from "What's the point of (something)"?

Both point and sense can be used in your example, but they mean slightly different things. Point here refers to purpose or usefulness and sense refers to good judgment.

You can use "what's the sense" as well as "where's the sense" and "what's the point".

They have similar meanings. "Where's the sense" and "what's the sense" are completely interchangeable.

I think it's fair to say that "what's the point" can be used wherever "what's the sense" can, but not vice versa. For example:

I can't be bothered. What's the point?

You can't substitute "what's the sense" in there.

As Jasper says, point refers to the usefulness of something, whereas sense refers to how sensible it is.

It seems to me that the question you're asking is the difference between "sense" and "point" more so than whether the question is grammatically correct. If I'm correct in that assumption, I would think that use of one or the other would hinge on what you want out of the answer.

I would say, "what's the sense" if I'm asking for a broader statement or a discussion, but I would use "what's the point" if I am intending the question to be rhetorical (and possibly, a bit sarcastic).

I would use "sense" in the stated question, particularly if you want an answer, and potentially an ongoing discussion such as: "This public space has been developed as part of the beautification of this area, and the committee feels that admission of food, beverages, and loitering does not add to the ambience we are seeking to create."

For many, the two are interchangeable, though, and could be rhetorical, depending on inflection and/or context.

(Personally, I would ask "what's the sense in" rather than "what's the sense of," but I think that's a personal preference rather than an issue of correct/incorrect.)