As a first-order approximation:

• Contrast-detection autofocus is more accurate and more flexible
• Phase-detection autofocus is much faster
• Contrast-detection always requires "hunting" back and forth to find the best focus; in ideal conditions phase-detect moves certainly to the right point
• Contrast-detection which uses the main sensor is likely to hurt battery life
• And as a practical matter, it means using the LCD screen or an EVF instead of the optical viewfinder

If your phase-detect system (camera and lens) are carefully micro-adjusted for the aperture, focal length, and focus distance you are using, results will be perfect.* Otherwise, phase detect results are likely to not be perfect — but they're generally good enough for 99.9% of people in 99% of cases. And the big speed advantage is real. So, that's why people count it as desirable.

As algorithms improve, and faster hardware comes to newer cameras, the speed advantage will come down, but it's hard to get around the hunting. The state of the art right now is hybrid modes, which use on-sensor phase-detect areas to get into the ballpark very quickly and then contrast-detect to fine-tune. I expect that eventually, pure phase-detect will become rare. But then, ten years ago, I thought we'd all be using Lytro-based cameras by now. It's hard to make predictions, especially about the future. :)

There is no inherent reason for exposure to be different. However, it is likely that as a side effect, your camera is also metering differently, using the main sensor instead of separate metering sensors. That probably accounts for the difference you are seeing in exposure, but I'd count that as a quirk of your particular camera model rather than something inherent. (And in any case, which exposure is chosen is a matter of preference, not something necessarily better or worse.)

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* In fact, using contrast-detection mode is my favorite way to do micro-adjustment — but unfortunately, Canon doesn't provide that as a user-accessible feature for your camera.