Sorry... that might as well be Greek....lol.
It's a bad deal for the average consumer, especially ones that can't afford to pay higher cable rates & fees.
Under a horizontal integration merger, two companies that are direct competitors merge, thus reducing the competition available. This type of a merger moves the industry closer to a monopoly. Had Verizon merged with AT&T, for example, that would have been a horizontal merger and would have limited competition.
In a vertical integration merger, two companies that are NOT in direct competition with each other merge, but the two companies are in the same supply chain. That's what happened in this case. Proving that there's less competition is extremely difficult in this type of a merger because the two companies merging are in different markets/industries. In the AT&T and Time Warner case generally, AT&T is a telecom provider -- they provide the transport while Time Warner does not provide transport, but rather they provide content.
I hope that helps.
EDIT: Added comment about the supply chain.