I have 4 oscillating tools. A corded Fein, a corded HF, a cordless Ryobi, and a cordless 12V Milwaukee. The HF and cordless Ryobi are similar and not very useful unless you are cutting drywall or you have a lot of time. They work but slowly. The Milwaukee and Fein are also similar. I actually did a timed test cutting a small board and their times were near identical (and about half the time for the HF or Ryobi). Both also have similar tool less blade change systems and both take the same blades. Bosch and Fein use a special type blade, I think it is called star lock, that others do not use. My Fein is older so it uses the same type blades the other tools use. Star lock blades are backwards compatible but I do not think the newere Bosch and Fein tools can use the older blades. You may want to check to be sure that the Bosch you are considering does not lock you into these special blades only available from Fein and Bosch, I believe.
I highly recommend the cordless Milwaukee. There is an 18V model too that is similar. At least the 12V is similar to the Dremel in price, even if you need the battery and charger. Cordless oscillating tools are handier. I reach for the Milwaukee virtually every time I need a tool like this. Nothing wrong with the Fein but it is just handier to use a cordless that cuts just as fast. I like Fein and Bosch blades but I've been using Milwaukee's blades too and they work quite well. I think I may have also used a dremel blade and it worked fine. If you need to cut metal, the carbide blades last a LOT longer than the bimetallic, but they are more expensive too. Blades for cutting wood last longer and I don't think the brand matters as much. Not sure about grout blades.