Are you saying that you would have no control over what you did with the money or that HD would have no control over it?

Oops. I meant to suggest that HD would not have any say in what you did with the money. Maybe just to say "brought to you by Home Depot" on your publications and stuff.

but if the latter what would be the problem?

Well, that's the crux of the debate. What about credibility? What about the mere knowledge that the company funding you is working contrary to your goals everyday? To make it a more black-and-white example, what if some multi-national fishing company wanted to give money to Greenpeace (and Greenpeace REALLY needed the money)? Does the money become tainted just because the donor might be using it to improve (or flat out be deceptive about) their corporate image? I sure don't know the answers to these questions, but.