Mr. Weatherby appears on the scene during the closure of the John Galt Line. He's the "man from Washington" sent to keep an eye on things at Taggart Transcontinental. Mr. Weatherby is in many respects a typical Washington looter. He's a fan of backdoor, shady dealing. He never comes out and says exactly what he means. He tows the party line and spouts off lots of rhetoric, and he enjoys manipulating people. So, your basic looter, really. Here's a scene where he threatens James with kindness. Seriously, it's a skill.
"It's the interests of the country as a whole that he's got to consider above everything else." Taggart sat up; of all the danger signs he knew, this line of talk was the worst. "Nobody can deny it, Jim, that Wesley feels a high regard for you as an enlightened businessman...and one of his closest friends." Taggart's eyes shot to him swiftly; this was still worse. (18.104.22.168)
Interestingly, Dagny actually seeks Weatherby out when she returns from her Woodstock vacation in the wake of the Taggart Tunnel disaster. She refuses to work with Wesley Mouch, recalling how he betrayed Hank, and tells Weatherby to keep everyone else in Washington off her back. Weatherby seems excited by the prospect of controlling Dagny, but that of course doesn't pan out for him.
In fact, the proper Mr. Weatherby seems to be the least influential of the looters by the end of the novel. He's been supplanted by thugs like Cuffy Meigs. Weatherby's rise and fall says a lot about how the looters' system of government changed over time.