Plus, most climbers have a series of rapid warm-up and cool-down cycles -- climb, stop, belay, climb, stop, etc. -- which doesn't get the heart rate going the same way that an extended aerobic workout does.
Your climbing will tend to improve if instead of climbing a few hard routes, you climb a bunch of routes about one full grade below your maximum ability. Bonus points if you can score a belayer that doesn't mind the boring end of the rope for an entire evening. As your muscles become fatigued, your body will start trying to get lazy and sloppy, and that's where your technique will actually improve. Straining on a really hard route doesn't do anything good for technique / body position / etc.
(Former climbing instructor from Livermore. I really miss it, can you tell?)
The soreness I get after climbing is the same kind of soreness I used to get during the first two weeks of a weight lifting routine. It is not tendon damage or severe muscle damage because it doesn't last more than a day.