Many people are going on about how the merger of Comcast and Time Warner will enable them to create a lane for the internet where wealthier individuals or companies can pay more for a 'faster lane.' What I'm wondering is why wouldn't current antitrust laws prevent this from happening.
So you're correct in stating that competition is at the root of the net neutrality debate. If there were multiple options for internet providers, then they would be able to compete to bring services to customers in a manner that customers wanted; if Comcast couldn't reach a deal to get Netflix at a reasonable speed, then customers could switch to Time Warner or Google fiber who would have every incentive to do a better job at delivering the services they want. The problem is that currently most consumers don't have many options for internet. If they don't like the way Time Warner does business they're still stuck with Time Warner. Thus there is a movement (net neutrality) to regulate the way Time Warner provides internet and tries to charge money from content providers, since Time Warner is a monopoly.
However, the Clayton Antitrust Act will only help if there are multiple competing firms trying to combine to form a monopoly. In the case you mentioned, Time Warner and Comcast, as Tyler Cowhen puts it "serve separate districts." One might own a cable network in Philadelphia while another would own one in New York. The fact that there is a competitor in New York doesn't offer a viable alternative for a customer in Philadelphia to switch to. That customer can't switch between Time Warner and Comcast, so keeping them from merging wouldn't keep their cable company from doing things unfriendly to consumers, even if they were separate. On the other hand, there is an argument to be made that a single monolithic company with access to many individual cable networks would have too much bargaining power against smaller content providers (like Netflix). That being said, any undue power would dissolve as soon as there was a healthy competitive environment on the various local cable markets. Cable providers would then have to fight each for consumers business.
In my opinion, there is a way help to encourage healthy competition in local cable markets and make the net neutrality issue less important; tear down the barriers to entry in local cable markets. If local governments stop making it so difficult for new entry into local cable markets, competition can solve the problem of 'fast lanes' that don't give consumers what they want.