I did a search and found nothing comparing the two.

What are the differences and advantages?

Thanks,
Cameron

Phantom.  Better flash supression than either.  Otherwise go for the A2 for it's reduced length compared to the Izzy.

Which brake has the best control keeping the barrel from rising?

Quoted: I did a search and found nothing comparing the two. What are the differences and advantages? Thanks, Cameron

The A2  or A1 birdcage is a Flash suppressor.  Most people on this fourm feel that the Vortex and the Phantom do a better job,  but I have found the Birdcage to be a very good flash suppressor.

The Izzy is not a flash suppressor.  The Izzy is not even a muzzle break.  It was designed by Bushmaster during the Assult Weapons Ban.  It satifisfied the ATF because it did not reduce flash, and it satisfied some of us gun buyers who did not want a "naked" muzzle.  It gave a ban compliant gun a pre-ban look.

Sence the AWB ended, Bushmaster started boring out the Izzy break to make it work more like a real flash suppressor.  But the last time I spoke with Bushmaster about it they said it was still more of a compensator and not a flash suppressor and not a muzzle break.

Quoted: ....  Most people on this fourm feel that the Vortex and the Phantom do a better job,  but I have found the Birdcage to be a very good flash suppressor.

No Tweak and I dont' "Feel" it's better.  We know they are better after using them at night with a variety of ammo.    For example Wolf fired through an A2 FS produces MUCH more flash than the same ammo through a Phantom or Vortex.

The Izzy is not a flash suppressor.  The Izzy is not even a muzzle break

.  
Incorrect on both counts.

There is an Izzy Flash Supressor (doesn't do a good job, but it's a heck of alot better than any muzzle brake or even a bare muzzle).  Izzy Flash Supressor

There is also an Izzy Muzzle brake (again doesn't do a great job, but it looks better than the alternatives and doesn't have that annoying side blast). Izzy Brake

Quoted: Which brake has the best control keeping the barrel from rising?

The A2 birdcage and the Izzy do about an equal job of controlling muzzle rise.
Which is some, but not much.

In my experience the best muzzle break for controling muzzle rise is the Bushmaster AK74 break.
In fact it works so good, I removed it from my rifle.
It made my AR15 feel like I was shooting a very loud .22 LR.  

I was impressed at first, but I wanted my rifle to have some Kick, so I removed it.

I have an A2 birdcage on that rifle now.

Quoted: Quoted: ....  Most people on this fourm feel that the Vortex and the Phantom do a better job,  but I have found the Birdcage to be a very good flash suppressor. No Tweak and I dont' "Feel" it's better.  We know they are better after using them at night with a variety of ammo.    For example Wolf fired through an A2 FS produces MUCH more flash than the same ammo through a Phantom or Vortex. The Izzy is not a flash suppressor.  The Izzy is not even a muzzle break .   Incorrect on both counts. There is an Izzy Flash Supressor (doesn't do a good job, but it's a heck of alot better than any muzzle brake or even a bare muzzle).  Izzy Flash Supressor There is also an Izzy Muzzle brake (again doesn't do a great job, but it looks better than the alternatives and doesn't have that annoying side blast). Izzy Brake

OK, you found the Phantom to be better.  My point was that the A2 is still pretty good flash suppressor.  Not better.

I am well aware that Bushmaster is selling an Izzy "break" and a Izzy "flash suppressor".
But the fact is the Izzy break is a terrible break and the Izzy flash suppressor does a pretty bad job too.
That is why I said the Izzy is neither a break nor a flash suppressor.