The difference is lobbying is soft influence, meaning if the politician's constituents care about an issue, lobbying and donations won't buy policy. What they do buy is sway in cases where the electorate isn't paying attention (essentially establishing a default position on an issue).
In other words, the decision process for a politician goes something like:
1. Do voters in my district care[1]/will they notice?
2. What will best attract continued campaign funding?
3. What is best for my district?
4. What is best for the country?
[1] The key here is whether voters care enough about an issue for it to influence the representative's reelection. If not, they can move on to step two.
Voters do come first if they care enough/make enough noise (e.g., the SOPA protests). The problem is it's impossible for voters to pay attention to every issue and coordinate their voices; that's why we have a republic in the first place.
National opinion polls may have been against the bank bailouts, but national public opinion is not what really matters to a politician. Representatives are accountable only to their own districts. Of course, this is why we see pork barrel spending even though everyone hates it. People usually like their rep and hate most of the others around the country. If the representative can pump enough cash into their district, voters in their district will overlook a voting record they mildly disagree with. Many representatives probably saw the bank bailouts as some combination of necessary for the country and a chance to appease their financial backers, offering a favorable trade-off in terms of displeasing their constituents.
Anyways, what would you consider the lobbyist mechanism and campaign donations if not an authorized de-facto form of bribery?