So move will fire in passing, while attack move will stop.
Why would I want them to stop? Would it not be advantageous to keep moving?
I guess what I'm asking is why pick one or there other, there has to be a reason to make the distinction.
This never really got answered. "Stop" is not actually really quite accurate. "Stay within range of targets I come into firing range of" is a better descriptor. So if your ship kills the thing in passing, then fine, it just keeps going. If the thing it is attacking doesn't move, then sure, it stops. If the thing it is passing is itself headed somewhere else, however, then your ships will break off to engage.
Sometimes you want that to happen, sometimes not. For instance, if the AI is sending in a huge mass of ships that are all splitting up like crazy, you might attack-move into the middle of them in order to have your guys split up and give chase (though honestly, if this is your own planet, Free Roaming Defender mode is VASTLY better for this purpose).
But let's say you are strafing a big blob of enemy ships and you want to get past them without having some of your ships stop and some not stop (different ships with different ranges, recall). If you attack-move, then the ships that come into range will not only fire, they will stop. The shorter-range ships won't fire if they are out of range, and will keep moving. Worse, if some of the enemy ships break away, some of your ships that are within range of them will, too.
Well, I say "worse," but that actually could be perfect. If you are attacking a cluster of enemy units that is not already mobile, and you are on an enemy planet, then FRD mode is a terrible idea -- really, FRD on enemy planets is basically "suicide mode" for your ships.
Anyway, sending in your ball of ships in attack-move mode against that cluster of guys can be a great way to not only attack the ball, but catch any stray enemy ships that break away by having your ships pursue them as needed. This is another case where group-move is a really bad option, because you want to have your fast ships able to pursue the breakaway ships, while your slower ships are going to be firing from range anyway, so they don't really need to stay together.
Anyway, attack-move can also be a negative thing to use if you want to keep orderly ranks of ships. Aka, let's say you want to have some certain ships of yours at one range, while others flank and draw away the enemy, and then a third group moves in. If you want the ships to definitely completely listen to your orders and not stray around and exert a bit too much self-determination, then a regular move is definitely the order of the day. Then once your basic trap is sprung on the enemy, doing an attack-move a short distance away to have your guys do cleanup and catch anyone who is escaping is kind of the perfect followup.
So really it is extremely contextual. I don't really use group move because I use waypoints and staging areas instead, and that's just how I got used to playing (from far before there was group move). But group move certainly is very useful. Attack-move is what I use 80% of the time on enemy planets that I am actually attacking, while the other 20% of the time I'm doing more precise maneuvers that require just regular moves. If I'm passing through an enemy planet and not intending to try to take the whole thing out, then it's definitely move 100% of the time. If I'm on my own planet and there's no one around, move is just as good as anything else. If I'm defending my own planet, then FRD.
Hope that helps.