To a degree, choosing Curves or Levels to make your tonal adjustments in Photoshop is much like choosing Coke or Pepsi to quench your thirst - it's a matter of taste. (Although neither Curves nor Levels go well with rum!)
Needless to say, Levels and Curves are the "heavyweights" of Photoshop - if you really want to be able to edit your photos, you need to be able to use one of them. They have some similarities, and some differences, and I'm going to discuss those briefly here.
I'll begin by saying that Elements users don't get this choice. Yes, there's a primitive Curves function buried in the Enhance menu in Elements, but the control it gives you is minimal, and more importantly, it's not available as an adjustment layer. Any functions that aren't available as adjustment layers should be avoided where possible, because they can't be re-adjusted, and they can't be masked. Elements peeps, you need to embrace Levels, and in some cases, Brightness/Contrast. Both of those are available via the little half-black-half-white circle icon in your Layers palette.
Ok, back to Curves vs Levels ...
No difference: the ends
At the ends of the tonal range, they are exactly the same. If you move the black slider to the right in Levels, the result is identical to moving the lower left endpoint of Curves to the right along the bottom of the grid. Likewise, moving the white slider in Levels is the same as moving the upper (highlight) endpoint of the curve along the top of the grid.
In newer versions of Photoshop, Curves also has sliders at the bottom, which makes Curves even more like Levels in this regard.
At first glance, there appears to be no difference whether you move the middle (gray) slider in Levels, or make a point in the middle of the Curve and move it around - both methods lighten or darken your midtones. For most subtle adjustments, either function will do a satisfactory job.
But if you start to push the adjustment further (eg if you have a very dark photo that you're trying to lighten), there are significant two differences between Levels and Curves:
(a) Aggressive midtone adjustments in Levels tend to reduce saturation of the colours in your photo; whereas aggressive Curves midtone adjustments tend to add saturation. For any given photo, either of these might be good or bad situations for you. Sometimes I lighten aggressively with Levels, then decide that I'd like some more saturation to go with my lightening, so I scrap that layer and use Curves instead. You'll need to decide on a photo-by-photo basis.
(b) Curves midtone adjustments, if strong enough, can cause clipping in the highlights and shadows. Levels midtone adjustments, no matter how strong, can never clip. This might seem like a tick in the box for Levels, but again, it's something you need to decide per photo.
Note: When you make a new midtone point in Curves, you might wonder whether you're supposed to push it up-down, or left-right, or diagonally. I can assure you it doesn't matter in the slightest. The result is exactly the same. Likewise, the point you make at the beginning doesn't need to be exactly in the middle of the curve.
Difference: multiple points
This is the obvious one, and I think it's the reason that some people think that Curves are better than Levels. In Curves, you can make multiple midtone points, and adjust different parts of the tonal range specifically.
I do a LOT of editing, and I probably use multiple-point Curve adjustments only once out of every twenty or thirty images I edit. It's not that big a deal, and frankly, you need to be aware of the drawbacks associated with multiple-point curves. (I wrote a little about them here.) Every action has an equal and opposite reaction, and for every bit of contrast you add to a tonal range, there's an equal and opposite diminution of contrast somewhere else.
No further difference: colour channels
Apart from the great power you can wield just by controlling the RGB channel, there's even more power to be found by adjusting the individual channels. Here, you can control the colour balance - eg a bit more red here, or a bit less green there, or whatever.
On the individual channels, the characteristics of Levels vs Curves are much the same as above. The ends behave exactly the same, and middles behave as I've already described. Generally speaking, you won't be making enormous adjustments to individual colour channels, so the real-world differences are negligible.
Choose your weapon
When it comes down to it, use the one you feel comfortable with. As your skills improve, you might like to master both of them, but as you're starting out, choose the one that makes sense to you, and go with it. Some people (of whom I am one) like the simplicity of Levels, but others find the midtone behaviour of Levels unintuitive, and prefer the up-and-down approach of Curves.