I've played around with both lenses a little (I own the 70-300IS and I had a brief loan of the 55-250) and my conclusion is that the 70-300IS is the better lens, but the 55-250IS is also very good.
The 55-250IS is an EF-S lens meaning that it's designed to cover an APS-C sesor, while the 70-300IS is a full frame lens. That means that when you use it on an APS-C camera you're using the center part of the image circle, where image quality is best.
The most noticable shortcoming of the 55-250IS is that there is clearly visible vignetting when you shoot it wide open. The vignetting goes way when you close down a stop or so, but with a telephoto that's only f5.6 wide open (at 250mm), you might want to shoot wide open when you can. I've attached a shot taken wide open (f5.6) at 250mm so you can see the effect.
With the 70-300IS on an APS-C camera, you don't see any noticable vignetting wide open at 300mm. You probably would on a full frame camera of course, but with APS-C you're cropping off the edges of that full frame image.
The 70-300IS is also longer of course, which is another advantage. It's not as wide, but with a telephoto zoom I find that I'm usually working at the long end of the range rather than the short end.
So my conclusion is that I can certainly recommend the 55-250IS as long as you're aware of the vignetting issue. It's sharp, the IS works well and the build quality is acceptable. The 70-300IS has somwhat higher build quality. I'd go for the 70-300IS if the price is acceptable to you. At half the price, the 55-250IS is quite a bargain and probably the better value, but I think the 70-300IS is the better lens. Note that the 55-250IS is still hard to find in stock at most vendors.
As far as sharpness goes, there's not a great difference between the lenses. Both are quite sharp, even at their long end.
I didn't get the chance to test the 55-250IS with the Tamron 1.4x TC, but I suspect it's not a combination you'd want to use much. You'll lose AF, which is a big loss, and you'll be at f8 wide open which is slow. Remember that the 70-300 is approximately the eqivalent of the 55-250 with a 1.2x TC on it - but with an f5.6 aperture, full autofocus and good image quality.