I know...nobody wants to feel like they've got one. more. product. that they absolutely HAVE to have. Can't one thing just do it all?! The answer is: maybe.
When it comes to the skin under your eyes vs the skin on your face, the two areas can have very different needs. The under eyes, for example, often have shadows that need correcting & brightening. The skin here also undergoes a lot of movement, so the products used here tend to crease easier. Finally, because the under eye is sometimes sunken in, it can help make this area look less shadowed if you use a slightly lighter concealer.
When it comes to the skin on your face, you are often covering pimples, or other discolorations like redness and darkness. If you use a lighter concealer on these areas, it will have the opposite effect of what you want—pimples will look more prominent, and other imperfections may be highlighted. You really need to make sure the concealer on your face matches your skin tone exactly. Also, because the skin is thicker and moves less than the skin under your eye, I do believe you can use thicker, cream based concealers with more success on the face.
So DO you need to get two different concealers? That decision is up to you and whether or not you feel you need the separate benefits two types of concealers can offer you. I myself love to use the click-pen style under eye concealers (like this one from Ellis Faas) on my clients who need a little color correction and brightening under the eyes. I then use a separate cream concealer on their face. On myself though? I find my under eyes cover up just fine using the same concealer I use one my face (which is L'Oreal Full Wear, btw), so I use it for both! It is indeed a time saver, but when it comes to anything in makeup, the bottom line is that you've got to assess your own needs and make the choice that's right for you.
PS. If you suffer from acne and have been wondering how to best cover up your breakouts/scarring...I just released a tutorial for that, so check it out!