Although I agree with Martha and the caller that the literal interpretation of being humbled vs being honored by an award makes saying "I'm humbled" seem odd, I disagree that saying "I'm honored" sounds more modest. To me, saying that you're honored actually sounds presumptuous, as if you were expecting the award all along. Humbled implies a feeling of genuine unworthiness. My best guess is this is an effect akin to double negatives: the logical sense is that you end up with a positive statement, but the effect is often is to emphasize the negativeness. So hearing the word "humbled" emphasizes humility, while hearing "honored" emphasizes the receipt of the award. But you're right that the final impression will be modified by how the speaker delivers the remark and what they say to accompany it!
Totally agree with Robert. I'm with Martha and Ann on "I'm humbled". You are humbled when you're brought low, not when you win an election or receive an award.
I find it rather insulting when a "winner" says they're "humbled".
The googosphere is full of comments over 'why are people humbled by an award?'