So you harpoon a comet with a velocity relative to you of 10 km/s. You have 1000km of tether. That means that if you didn't apply any braking, you'd have just 100s before your tether ran out. With little bit of physics, I find that the probe would need to accelerate at 50m/s^2 (over 5g) to match velocity with the comet before running out of tether (which takes 200s because now it is accelerating). So the tether and the harpoon need to withstand a tension of 5 times the earth-weight of the probe, without breaking or pulling the harpoon out of the comet. Now assume the probe is 1000kg. (For simplicity I'll ignore the mass of the tether and the rotation of the tether reel, although that would probably be a deal breaker too.) Force = 1000kg x 50m/s^2 = 50000N. Distance acted over = 1000km = 10^6 m, so work = 5x10^4 x 10^6 = 5 x 10^10 J. It is 200 s before the tether runs out, so power = 5x10^10/200 = 2.5x10^8W = 250 MW. That power has to be stored and/or dissipated, and you have at best 1000kg with which to do it.
It all gets very much easier if your relative velocity with the comet is much lower, but then you're not gaining much, and intercepting a comet with only few km/s relative velocity is very hard in itself.
It is a pretty idea, but I can't see it working with anything vaguely like current technology.
Does anyone care to poke holes in my reasoning?