Proxxon only makes 4 versions of their rotary tool, and none are battery powered.
They have 2x mains version that go up to 20k rpm, and they have 2x 12V versions that go up to 15k rpm. Note that if you buy the 12V version, you shall need to buy the PSU separately. The PSU is labelled as DC output, but it's actually non sinusoidal AC with a bit of DC off set. With these 12V tools, you can run them from DC/battery, but you don't get any speed control.
The Dremel tools have much higher speed, 30-35k rpm, and more torque than the Proxxon tools. Maybe twice as much, roughly. The Proxxon tools are lower in noise and vibration, though.
You can use the same bits in either, but you have to buy a chuck or the collet set. (Proxxons come with full set of collets OR a chuck adapter. Dremel usually just come with a single 1/8" collet. Also, the Proxxon uses a skinnier and longer mandrel on their bits, and the Dremel won't be able to chuck the Proxxon bits as short unless you cut down the length of the shaft. The Proxxon spindle has a deeper hole in it, I guess you would say.
I use Proxxon rotary tools more than Dremel. Noise is probably the main factor. If it takes twice as long, but it is quiet and smooth, that's usually a good tradeoff for me. I keep telling myself that I am going to buy a proper router, and I probably am, soon. But these rotary tools can do quite a lot with some patience.
One weak point in the Proxxon is the ball bearing that goes around the spindle. The tool is very skinny where you hold it, but they use a really thin ball bearing in there, and I've worn them out, twice. If you replace it as soon as it starts making a funny noise, the tool will keep trucking, but the motor will suffer irreparable damage if you let it go too long. Cost is like 8.00 from McCarr Master.