Image quality. The wider the range of focal lengths on one lens, the more design compromises are made and the more correction must be applied deal with things like geometric distortion, chromatic aberration, and light fall off in the corners.
Aperture. Even though the EF-S 18-200mm f/3.5-5.6 has the same maximum aperture of f/5.6 as the EF-S 55-250mm f/4-5.6 IS II at each lenses respective maximum focal length, the 18-200 is slower for most of the range they share in common. The 18-200 is at f/4 by 28mm, f/4.5 at 45mm, f/5 at 55mm, and f/5.6 from 80mm on up. The 55-250, on the other hand, doesn't reach f/4.5 until 74mm, f/5 until 96mm, and f/5.6 until 154mm.
Price. Depending on where you buy, the kit with the EF-S 18-55mm f/3.5-5.6 IS II rarely costs much more than the body only for the cameras it is offered with. I have seen occasions, usually during camera+lens rebate promotions where the kit was actually cheaper than the body only! The current difference at amazon.com for a T4i body is $16 less than the kit. So the 18-55 costs very little. The EF-S 55-250mm f/4-5.6 IS II goes for about $300 in U.S. stores, but you can get it from amazon.com for $174. The EF-S 18-200mm f/3.5-5.6 IS sells for around $700 in the stores and $569 on amazon.com. The 18-55 & 55-250 will run you around $190 more than a T4i body. That's $379 cheaper than the 18-200. The Sigma Sigma 18-250mm f/3.5-6.3 DC OS HSM IF is even slower than the Canon 18-200, costs $349 and performs about the same or worse than the Canon 18-200.
The-Digital-Picture reviews the EF-S 18-200mm f/3.5-5.6 IS lens here, the Sigma 18-250mm f/3.5-6.3 DC OS HSM here, and the EF-S 55-250mm f/4-5.6 IS II here.
DxO Mark has all the quantification you could want of theses three lenses.
Here's a screen grab comparing the Canon 18-200, Canon 55-250, and Sigma 18-250. The charts show sharpness at maximum focal length and aperture of each lens. Green is sharper, yellow is in between, and red is less sharp.