Zoom for APS-C Camera, Canon 70-300mm, Tamron 70-300mm or????

For absolute image quality, the 70-200 f4 IS is the easy winner. In your stated price range, the non-IS version of the same lens wins, albeit without IS (but note that this is partially compensated for b/c it is a stop faster than the competition at the long end -- where you need it most).

That said, while I do not have the 55-250, everything I have read from lens tests to comments posted here, suggests that it is a real overachiever at its price point.

For myself, I saved and waited, and got the 70-200 f4 IS (refurb from Canon, can't tell it from new, and it is a great lens). That said, you question was "best bang for the buck." Unless you are going to make yourself crazy doing pixel-level comparisons with the 70-200, I suspect that the "bang for the buck" answer is to get the 55-250.
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WLW