And the answer is ...
Rooting allows you to access system files and run apps that require root. So you can freeze bloatware (freeze it first, then if freezing a particular app causes a problem you can unfreeze it, rather than having to reflash the ROM - if the phone works for a month, delete the frozen apps), and you can run things you couldn't run otherwise (like
Xposed Installer, which has many modules you might want to run). If you're familiar with Linux, Google left out su, rooting puts it back, so you can su. (It works differently in Android, you don't su an app, you just su, then you're running # in the shell.)
The disadcantages are a) it voids the warranty on any phone but an HTC, and b) you
can wipe out the entire phone with a single shell command, or mess the phone up in a lot of other ways if you play with things you don't understand. (Learning how Linux works isn't that difficult - millions of people run it.)
It also uses a tiny bit of storage - for the su file, for a busybox file (Google left out some commands, busybox brings them back) and the superuser app. (That's so that you have control over which app gets root. Otherwise a virus could gain root, steal all your stored logins and passwords and sent them to some hacker.) Not much space - su and busybox use about 1MB, SuperSU (the superuser app I use) uses 3MB.
My viewpoint is that if you have no need for rooting, don't. If you need to do something that needs rooting, don't be afraid to root, just be aware that you'll have to unroot if the phone needs warranty service. (There's a difference between "want" and "need". If you're out of storage, you need to get rid of bloatware. If you have 16GB free, you may want to get rid of it, you there's no need to.)