Russell, as a mathematician, was principally interested in logic as the foundation of deductive systems in mathematics. He also, as a philosopher, was interested in set theory and paradoxes such as Russell's Paradox: Is the set of all sets a member of itself? If you go this route, consider, too, Godel's incompleteness theorem. Russell also wrote some very accessible books on philosophy and the history of philosophy. Great reads by A Nobel Laureate in literature, and biased--all philosophy culminates in him.
Courant's book is focused on straight-up mathematical topics, spanning the range of topics that a math major would study.