Tae Kwon Do and Karate ( of the Shorin and Shotokan lineages ) share common roots, so it's difficult to talk about individual advantages as they're so similar in some ways. In my experience, Tae Kwon Do will teach you more and better kicks than Karate, and will also get you into sparring faster. Depending on the specific lineage, Karate can give you better hand techniques and better form ( I was with a branch of JKA Shotokan, little sparring, but absolute dedication to perfection of technique ).
There are other considerations though. Many instructors don't like their students crosstraining in other arts. Furthermore, the instructors may want identical techniques done very different ways. The TKD school ( if ITF ) may want you to emphasize the sine wave in stepping techniques, while the Karate school may want you to keep your body absolutely level stepping. By advancing in one style you may actually regress in the other.