While you can make animations in photoshop, it's really more for editing photos and painting. Flash CS6 is designed to create animations. Once you are comfortable with Flash CS6 you will be able to create animations easier an quicker. While you can create animation in photoshop, it's not as easy. Photoshop is raster which means everything created in photoshop is made up of pixels, everything created in Flash CS6 is vector based. Now you can create psd files in photoshop and transfer them straight into Flash CS6 to animate the psd, just like you can in Adobe After Effects. When it boils down to it, it is how you can make any of adobe products work for you. You said you like creating anime, I would recommend working in Flash CS6 for animation. But I will also recommend doing some testing yourself and see what works best for you. That's the great thing about Adobe software they give you the freedom to create whatever you want with whatever Adobe software you use.
Now recently Adobe changed flash's name to Animte and they have added some great new tools to make animation even easier. You may want to look into upgrading.