I don't own a Fuji X100 and I have not tested the Canon G1-X, but I've talked to a few photographers who have.
The Fuji X100 has been a smashing success, both in quality and in sales. The Canon G1-X is supposed to be just as good, but for $400 less.
So which is better? Which should you buy?
The X100 looks much better, but looks should mean very, very little. You should not decide which camera to buy based on looks. You'd be surprised how many times I've heard that such-and-such camera looks "better" or more "professional" than another so that's why it was picked.
Perhaps the most important feature on a camera is the digital sensor. The X100 has a back lit APS-C sized sensor with 12.3 megapixels. The G1-X has a back lit almost-but-not-quite APS-C sized sensor with 14 megapixels. Both sensors are fine, but the X100's is better, and it shows (so I'm told) with higher ISOs.
To combat this, Canon programed the G1-X to apply a liberal amount of noise reduction in ISO's 1600 and higher. The result is poor sharpness and details in the higher ISO's using the Canon camera. To get around this you can save the image in RAW format, but you'll experience noticeably more noise when compared to the X100.
That may or may not be important to you, depending on if you plan to use the camera in low-light situations.
The X100 has a maximum aperture of f2 compared to f2.8 with the G1-X. This may not matter if you won't be using the camera in low-light situations. But the larger aperture plus the better high-ISO performance of the X100 could be significant, depending on exactly what you are photographing. You will have to decide for yourself if you need or want that extra performance.
One area where the G1-X is better is focal length. The X100 has a fixed (but very sharp) 35mm (equivalent) lens. The G1-X has a 28mm-112mm (equivalent) zoom lens that I understand is also sharp. That flexibility could be the difference from getting "the shot" and not getting it.
The last feature that is important on a digital camera is the software that the camera uses to process the images. If you are using RAW this is less important, but if you are saving in JPEG, this is very important. Without a direct comparison it is hard to tell which is better, but from what I've heard, the X100 has a slight edge.
Which camera has better image quality? Again, without a side-by-side study, it's difficult to know, but who (outside of product reviewers) does side-by-side studies? My guess is that there is not a significant difference in actual image quality in the low ISOs, and that both cameras produce quality images.
If you will be using the camera in low-light situations, the X100 is the better choice. Outside of that, I would give the G1-X a slight advantage for having flexibility in focal length.
The Fuji X100 retails for $1,200 while the Canon G1-X will retail for $800. The $400 difference seems like a lot of money for better low-light performance, unless a lot of my photography was done in low-light situations. My gut says that the G1-X is the better value and the choice I'd recommend (again, this is without having first-hand knowledge of the G1-X).
Another option to consider is the Pentax K-x DSLR (or the almost identical K-r), which has a sensor almost identical to the X100, but for less than the cost of a G1-X. Right now, and especially if you can find it discounted (shop around and you are sure to find it discounted somewhere), the K-x is the best value for APS-C sized digital cameras, period.