Both systems install to the hd. Installs on ps4 are faster because it doesn't install everything, I think some of the stuff is cached. Xbox one for most games installs the game and sometimes downloads the rest of the updates from the internet while it installs so it takes longer.
For my xb1 a game like forza horizon 2 took about 15-20 min. On the ps4 a similar size game installs in about 5-10 sometimes. Mlb 15 took forever to install since it had a lot of textures, same with gta v. Just my experience though. You really don't need 2tb of space unless you keep every single game on the system. I have about 12 disc games installed and I still have about 400+ GB left with maybe 3-4 games downloaded from the psn too also installed.
Both systems use the disc to read and validate some things.
As for pros and cons, at this point it only depends what games you prefer. If you prefer games like mlb, uncharted, bloodborne, infamous, go ps4. If you like halo, fable, forza, etc then xbox. I got the xbox 1 in the living room so I could stream media from my pc because at the time the ps4 could not do that.
Currently the ps4 got a media player so it can do the same thing now. For the xbox you need an external HD for extra space, for ps4, I replaced the internal hd with a 1tb drive.
It's really up to the type of games you prefer.