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The "Hadoken" you get from beating Ryu is actually about as good as a charged shot before charging, with its actual charged shot being stronger still. So in some cases (most notably against Chun Li) you can use this to great advantage. Even if I am remembering incorrectly (I don't have time to test it now to make sure), I do know you can quickly fire two plain Hadoken which each have damage similar to a charged shot, something the buster just can't do.
There is no difference in strength between the code Hadoken and the one you get from beating Ryu, aside from animation.
Anyway, the charged Hadoken does 1 more damage to Chun Li than the charged Buster. The charged Buster simply negates Chun Li's, Ryu's, Dhalsim's, Rose's and Akuma's fireballs but the charged Hadoken goes through them. Actually playing the game helps you learn a lot about how the weapons work.
There is one other difference, though perhaps it is too obvious to mention:
The cheat code Hadoken doesn't require an Energy meter (unless my game was running glitched); even if you drain the "meter" that appears by your Life Bar, you can still throw the Hadoken. It also requires the classic "Hadoken" control input of down, down/forward, forward+[shoot button]. Especially on a keyboard, this isn't always easy. So the regular hadoken still gives you what is, essentially, a metered but "better" buster.