Are there any safety differences between linking my debit/Visa card to a website and linking my bank account directly?

Asked by tinyfaery (44107) December 12th, 2008 from iPhone

I’ve been paying more and more of my bills online. My gas company will only let me pay online if I link my account, routing number and all, to their site. Now, I have no problem linking my debit card to sites, but for some reason linking my routing number seems less safe. Does anybody know if there is any difference? Which is more safe.