Scalability is one of the main assets that makes Illustrator a suitable tool for logo design. Apart from that, Illustrator is more advanced and powerful in designing (or dealing with) typography. The brush tool seems more powerful in Illustrator as it has more options. Photoshop is mainly used for photo-editing and retouch works. Sometimes you may need to use combination of both Photoshop and Illustrator for creating your designs. Photoshop can fulfill your needs but having Illustrator skills will definitely open more doors.

Hope this helps!