Many game consoles used to target the analog PAL and NTSC video standards used in televisions. These are old signal types, and they have very restricted properties to make TV manufacture simpler. Simply put, you have very strict resolution (you're stuck with the maximum the screen can show), and the image is refreshed at the set rate (your FPS rate).
NTSC is capable of 30 frames per second. (Technically, it's 60 hertz interlaced - it draws half of the horizontal lines of the screen, then the other half, hence 30 full frames per second). PAL is 25 FPS @ 50 Hz, though new televisions also support PAL60, which is 30 FPS @ 60 Hz as in NTSC. In newer console games in PAL region, many games simply don't support 50 Hz PAL any more to make development and porting easier.
The thing is, modern televisions are now ditching the old analog restrictions completely, and are essentially getting the complicated mode selections that have been the staple of computer monitors for ages.
A digital television broadcast can be in HD - fine, switch to a finer display mode. A blu-ray player is able to show stuff in HD - fine, switch to a finer display mode. A game console wants to render game at bigger framerate? ...fine, whatever! Switch to a higher refresh rate!
So this is basically the big deal. We're no longer constrained by the old, old, old, super-limited TV standards. We can treat TVs like they're just a funky type of computer monitors that - inexplicably enough - have built-in digital broadcast decoders.
There's a lot of debate about picture quality, but I find the whole debate extremely subjective so I don't really want to get into the things. Much of it deals with what kind of TVs/monitors you have and what you're accustomed to. I'm personally the sort of person who doesn't notice much when we're going up in quality, but I do notice when we go down in quality. (I started playing Destiny on Xbox One. Ooo, looks pretty... or something. Friends asked me to get it for Xbox 360 too. Guh, everything is blurry and jerky, but I can manage.)