I have been turning a fair amount of cottonwood lately. It turns well. Its not particularly hard, but not particularly soft either. It often has good figure, frequently has natural staining, often has darker heartwood depending on the size. It sands to a pretty good sheen naturally, takes a nice finish.
It does NOT seem to stain well...went down this experimental staining path a few times. I tried it with cottonwood each time, as a treecutter friend had a TON of cottonwood he needed to get rid of (too many wood piles, neighbors complaining), so I've been taking a lot of it. It just does not stain well. It finishes fine with standard finishes, but when it comes to stain the fibers of the wood, even with a pre-stain conditioner, just do not absorb the stain in any way resembling consistent or even. So I wouldn't stain cottonwood.
Aspen is similar, its not a harder wood, its a bit softer. With the aspen I've turned, it doesn't turn as well as the cottonwood. But, it does have this internal staining around the heartwood that is quite nice. Being slightly on the softer side its very easy to turn. I had a tough time getting it to take spar or poly finishes well. Don't know if it was just the particular species or not. IT was very, very dry wood. It took my more standard shellac + acks finish fine, but that's not a particularly durable finish.
I don't think I have any of the aspen pieces around, but here are some of the cottonwood pieces I've turned recently:
I like the cottonwood for its variation and figure. Almost every piece I have has some figure, most has some natural staining. Just makes for more interesting wood, I think. It takes a finish very nicely.