Moderators are the norm here in the UK. Easy to get and kinda useful considering the population density in areas where most hunt and also for range use.
I personally don't bother with one mostly out of aesthetic and balance considerations on my rifle, but I've shot plenty with them.
Advantages are:
Noise suppression. Most knock 30 odd decibels off the report, which relegates a centrefire discharge from painful to merely uncomfortable. Ear protection is still very much a good idea, but plenty don't bother for hunting, and for 1 or 2 shots, I can understand why.
The mod also seems to 'flatten' and disperse the sound better, which is an advantage for shooting groups of game. I hunt hinds in scotland with an unmoderated .270 with my friend who has a moderated gun of the same calibre. A quick follow up shot on a second beast in a group is a regular feature with hinds, but whilst they often stop and look back either way, with a moderated gun they often seem to have no idea where the shot came from in the first place and can even run towards the shooter sometimes.
The other advantage is in recoil reduction. I have no idea what the reduction is in real terms, but I'd say it makes a .270 kick like a .243. Certainly an advantage if trying to shoot multiple groups off a bench.
Muzzle flash is also much reduced, especially with shorter barrels or big powder charges, which can be helpful at dawn or dusk. Expect even more so if shooting vermin under NV.
Finally, a mod is a pretty good muzzle guard. You're unlikely to bugger your crown if you drop the rifle or scrape it, and the mod also reduces the risk of you accidentally plugging your barrel with snow or mud on a final crawl (as me how I know), which can be frustrating.
Velocities are unchanged and so is accuracy in most cases, just makes it easier to utilise the rifles performance if it isn't beating you up as much.
Disadvantages are weight, balance, length and aesthetic.
I hate shooting a moderated rifle off hand, nose heavy in most cases, and you do have to account for maybe 200-500g additional mass to lug around generally. Most also add around 2-6" to the over all rifle length, with my 22" barrel gun ending up 3" shorter than my friends 20" variant. This either necessitates a velocity loss, or a less wieldy rifle.
The final point is just personal preference, but I find them rather ugly, especially on a nice blue and walnut rifle.