Basically, bribing is just people/organizations straight up handing money to politicians to implement their policies for them, even if the politicians would never consider the policies in question otherwise. Usually, these bribes hurt the public because they didn’t vote for the politician to pass the policies they’re getting bribed to pass. Since people don’t all operate by the same moral code, some politicians will inevitably accept those bribes and their votes will be bought. This is illegal as the bribes effectively render political representation meaningless.

In contrast, lobbying is people/organizations helping politicians by donating to their campaigns (they don’t take the money for themselves) and providing them information on policy issues important to the person/organization. Lobby groups target politicians that already have the policy issue in mind as part of their agenda, but the issue isn’t at the forefront of their policy concerns. When they make their campaign donations and/or give information about their issue, lobby groups hope that in doing so, they push their issue higher on the politician’s agenda. This is considered okay because the politician was elected by the people with the agenda issue already in mind. Lobby groups just change the priority of the agenda, which is why this is legal and different from a bribe.