What exactly is the harm to user/computer of clicking a false pop-up?

It's all just software. It is not going to harm the machine. In some cases, there could be attempts to steal passwords.

This is is a good question because it really gets to the big disconnect on security that people don't understand. They will spend money on security scamware. They will load their machine up with 2 or 3 different legitimate, but ineffective, antivirus products. They are genuinely worried about their computer being vulnerable to hackers.

But the computer itself has been practically invulnerable since the day they first turned it on. It is the human that is vulnerable. People who do all of that in a vain effort to protect their computer are showing themselves to be even more vulnerable to social, rather than technological, hacks.

Generally, interacting with those scams, or even having been on the kind of website that is going to host them, is setting the user up for future scam attempts. The scammers will collect IP addresses and web tracking tokens, building a database of potential targets. It is the same "big data" that "legitimate" companies use to find potential customers. It is very effective.