My personal take is both do the same thing - in different ways. I personally prefer Corel Draw - as Adobe tries to have you use the keyboard with one hand, and the mouse with the other. Zooming is a primary example. It is easy to zoom in and out with just the mouse in draw - not so much with Adobe. The other side of the coin, Draw seems to be a 100% Vector based platform, while slowly adopting raster capabilities (drop shadows and transparencies) while Adobe seems to be primarily a Raster based editor that incorporates vector capabilities. This can be good or bad - depending on what your project requires. I think you will find more native support with hardware options (engravers, routers, vinyl cutters, tables etc) with Corel then you will with Adobe. However, my info on this is a bit dated, and things may have changed.
Bottom line, I find Corel easier for first time users. Adobe tends to have a strong tradition of "initiating" users before the basics are unlocked. However, Adobe has more third party support (especially for Photo Shop).