The main advantage of stainless is of course, it's more corrosion resistant. It's not 100% of course, but wander around in the rain for a day with a stainless barrel and leave it overnight 'cus you're out at camp, you might get away with it. Chrome-moly (CM) steel, not so much.
I understand there are Cerakote type coatings now which help externally, but I'm thoroughly unconvinced that they will give the same protection as true stainless, and they also offer no solution to the issue of moisture and therefore rust on the internal barrel surfaces and chamber, which is where I personally care the most about it. For my part I've discovered the first beginnings of surface rust in the chamber of my CM barreled .270 merely from making ready with a round whilst out in the rain then removing it later without firing. Total water contact time on the surface was less than 4 hours. A rusty crown and muzzle end of the barrel from leaving a moderator in place is also an absolute classic.
The next advantage of stainless is a double edged sword. It's (often) a bit softer than CM steel. The advantage here is that it's easier to machine, especially if cut rifling, which in theory gives a smoother finish, more precisely cut rifling and less tool chatter. Easier to lap too, potentially giving better accuracy, at least on paper. They also tend to foul less for this reason, subject of course to degree of finishing and quality.
I don't know of many 'match quality' barrels that aren't stainless, nor that many that aren't cut rifled.
The flipside of this is that they also don't last as long. I hear from competitive types at the range that stainless barrels tend to shoot pretty well for a longish time, then suddenly drop off a cliff (often when an entire chunk of a land flies off). I've seen this with my own eyes with a guy shooting F class starting his 20 round detail with a sub 0.5MOA rifle and ending with a 2MOA gun. On the other hand CM barrels tend to just get progressively and slowly worse over time, often shooting pretty well even when in appalling condition (I've seen old SMLEs and the like with a smooth bore till 6" past the chamber still shooting to the original 2MOA ish specification.