Due to the processing power of the PlayStation 4 Pro being used for more than upping the resolution, games on your standard high-definition TV will run smoother and faster than on a standard PlayStation 4.
Simply put: The power can be used for other things, and many of those things have a major impact.
Being able to lock the framerate of a game — the number of image frames that a game is able to render per second — is really meaningful. Ever play a game and, when a lot of stuff is happening on the screen, the game slows down? That's because it's "dropping frames" — the hardware is having a hard time processing all the information on the screen, and so it compensates by pulling away processing horsepower from other stuff (like how many frames are being rendered on screen per second, for instance). With the PlayStation 4 Pro, there's more than enough power to go around.
A variety of games on the PlayStation 4, pretty as they are, don't get rendered in 1080p — so-called "true" HD. Instead, they're slightly smaller (think: 900p). The PlayStation 4 "upscales" the games to 1080p instead of them being produced in 1080p. Essentially, the image is stretched as a result.
Not on the PS4 Pro, though — it can take your close-to-but-not-quite HD game and make sure it's running in full 1080p.