If you block out competition TKD and sport karate, and just focus on the two as combat arts, they're brothers under the skin—as you'd expect, given the roots of early TKD in Shotokan/Shudokan karate. They have way more in common than in opposition.
The main difference I see between the arts isn't technical, it's cultural: the culture of TKD is inextricably tied to its Olympic status (if you're primarily a tournament TKDist, you define TKD in terms of WTF-rules competition; if you're a combat TKD type, you spend a lot of your 'rhetorical' time denouncing Olympic TKD and showing how different what you do is from that), and to the associated battle between those who view the RoK directorate as TKD Central, vs. those who long for the freewheeling Kwan era, when every school was completely independent and you didn't worry about anything as long as you were down with your GM. Karate doesn't seem to be lumbered with anything like these internal conflicts and polarised allegiances. It seems much more open, less bloody-minded and less insecure about its historical sources, less inclined to insist on an heroic mythic history for itself.
Some of this comes from the difference between Korean and Japanese history and culture, some from the difference in the role of the two arts respectively on the world stage, but wherever it comes from, if you can just 'bracket' it and set it to one side, and look at the two arts in terms of potential SD applications, they appear extremely similar. As has been said before, zillions of times, in many different contexts, the important thing is who you train with, and what that person's take is on the content and application of the art...