A 20 will have less..
Recoil (faster follow up shots..think mtpl targets quickly)
Blast/noise (for indoor HD use this can be a big deal)
A 20 gives up payload weight. Not as many pellets of any given size. For a HD gun that hardly matters. 6 dbl ought pellets or 9, im betting Mr. Badguy csnt tell the difference
A 20 is generally lighter and faster handeling then its 12ga brother.
Slugs are avail for the 20ga just like a 12, so IF you feel the need to be able to shoot farther then Buckshot or need to punch thru thin cover. You can have some of these handy
For a close range HD/SD shotgun there is NOTHING wrong with going 20. Especially if you have a recoil shy member of the house that may have to use the gun. My Wife hates my Bennilli and my 590's (all in 12) but loves the 20ga 500 we have
As for brands.... I REALLY like Mossburgs. Top mounted safety (ambidextrous). Action bar release BEHIND the trigger guard (no reaching out in front of the trigger to pull it. Ive seen problems reacing out like that)
I dont know if they make a 590 in 20ga. If so id take that in a heartbeat. Metal parts instead of the plastic ones just make me happier. That being said the first dept i worked for had Moss 500's as patrol guns (1980's). They worked and worked and worked
Pistol grip.....id leave that for the movies. Hard to hit with em. A short length of pull stock is how i go on all my shotties
Just my 2c...