The difference between these two types of policy making is in the degree to which higher levels of government tell the lower levels what to do. In a vertical policy making structure, the higher levels of government are able to pretty much dictate to the lower levels. In a horizontal system, policy is made by a group of equals. This distinction is typically seen in federal systems such as the US or the EU.
For example, then, immigration policy in the EU has historically been made horizontally. Countries have worked together to formulate policy. Recently, though, there has been a move towards a more vertical model. In this model, the EU government tries to impose an immigration policy on the various member states.
The same thing happens in the US when the federal government takes more (or less) power over things the states do. Right now, for example, the House of Representatives is proposing to let the states have more say in how they will run Medicaid. This would be a much more horizontal system than the current vertical system.
Vertical versus horizontal policy-making is a product of the nature of the organization involved in the process. Horizontal policy-making exists when departments or individuals within an organization of similar rank within the organizational hierarchy work collaboratively towards a common outcome. It is a more democratic process, in that many parties can be involved in formulating policy. It can also involve a considerably more protracted policy-making process, as there might be a requirement for a consensus to agree on the final decision. Getting all parties to the process to agree can require many hours of arduous negotiations or discussions with the final product representing, potentially, the optimal policy choice of none of the affected parties. The inevitable consequence of a democratic process such as is involved in horizontal policy-making is that the final decision reflects so many perspectives that it constitutes a watered-down agreement. That, however, is the nature of a democratic process in which no one party dictates policy to the others.
In contrast to a horizontal policy-making process, a vertical process is more of a "top-down" arrangement, in which policy is dictated from above and compliance is expected at lower levels of the organizational hierarchy. A vertical policy-making process may involve input or recommendations from the lower rungs of the organizational ladder, and research and analysis provided by those lower rungs may be instrumental in the upper level's decision-making process, but it is the upper ranks that ultimately make the decision and dictate the terms of that decision down the ladder. As such, vertical processes may be far less deliberative than horizontal ones, but decisions or policies can be made much faster, as the head of the organization is empowered to dictate policy. If speed is an issue, then a vertical policy-making structure is far more efficient. If compromise is desired and/or required, than a horizontal process may be more appropriate.
Horizontal and vertical policy making differ in the level of responsibility shared at a given level or delegated from one level to the other. In horizontal policy making, all the constituents share responsibilities and are treated equally. Vertical policy making involves the higher authority (say federal government) delegating some responsibilities to the lower levels of government (say state or local governing bodies). The vertical policy making is akin to corporate structure, where the head office or top management makes a decision and it trickles down to the lower levels, who are free to make a regional level policy in accordance with the national policy. Horizontal policy making involves components at similar or the same hierarchical levels. The government is increasingly moving towards horizontal policy making as many of the decisions require inter-agency or departmental collaboration and also enable more cohesive governance.