The differnce between lobbying and bribery is the legality and ethics of them. Bribery is illegal. But, often times lobbying is too, it is just done in a legal way to get around it all. Lobbying can be just as crooked, in other words. It is offering large amounts of money in order to get one's way legally. Bribery is making a promise to give something, like money or a favor, to someone in return for their vote or support.
Lobbying is trying to sway a person through thoughts and ideas, trying to make them think the way one wants them to. But, lobbying can be done in an undercurrent way so that the person is really manipulated rather than being a free thinker. Bribery is always unethical and illegal. Both can be a big problem in government as you might imagine. Because outright bribery is illegal, many turn to lobbying.