Note that I'm talking about the total bandwidth available, not the speed of the connection to an end-user. The distinction is important: having more bandwidth means that you can send data to more users without sacrificing individual speed.
Peering arrangements between providers are the closest it should be possible to get to paying for a larger chunk of the last mile: because the server they are trying to talk to is fewer hops away, they will have lower pings, and possibly more responsive page-loading. In a situation where the connection of a network to the backbone is saturated, this appears to be the same as giving the peered provider's packets priority, since that connection is not saturated. This situation is somewhat distinct, however, being an issue of incompetence or third-party malice (the connection is either too small for normal usage or being attacked) instead of hardware designed to prioritize packets by source (which would be malice or greed on the part of the provider). This assumes a simple network topography, with a single connection to non-peered network, but it may be expanded to explain situations involving larger networks.
So you would have no artificial incentive to visit say the BBC rather then Huffington Post.
But, what internet prioritization is about, is that regardless of the speed of bandwith you have, some websites will be slower and some faster. Sure, if you have a 10mb and I have a 20mb then the slower sites will be even slower to you than me, and the faster sites not as fast as mine, but that is the point. Not that a user can access all sites faster or slower, but that all users can access some sites faster and some slower, regardless of the package they have.
So, if this was the case say in 1998, when Google was the newest and best thing, google.com would be so slow and Yahoo so fast that possibly people would just stick with yahoo.
I am sure it won't be difficult to find a conflict of interest given enough time and freedom for verizon to strategize. For a hypothetical example, what if verizon itself spawns a paypal like website using its own secure service for free for which others have to pay?