Vehicle patrol means greater speed and police visibility, while foot patrol means greater ability to notice small details that an officer would be likely to miss in a vehicle, even if driving slowly.
The advantages of vehicle patrol include speed, the ability to chase a suspect quickly if needed, and the ability to cover a much larger area geographically. As an added advantage, a vehicle patrol can arrest and transport a suspect without needing to call for backup. Disadvantages of vehicle patrol include a lack of engagement with the community and inability to see as much as an officer on foot would see.
The advantages of foot patrol include the ability to talk to and question members of the community, and to be hypervigilant. Disadvantages include the need to call for backup in a myriad of situations and a lower geographical area that can be covered when compared to an officer in a vehicle.
In terms of community relations, an officer on foot is infinitely more approachable than one in a vehicle. If an officer is on foot, he or she can be approached with any concerns that a member of the community may be experiencing.
In terms of police efficiency, however, an officer in a vehicle can respond to a problem more quickly if they receive notification of an incident unfolding nearby. The previously-mentioned issue of being able to transport apprehended suspects also comes into the argument about efficiency.
Ultimately, the choice of whether a vehicle patrol or a foot patrol will be better will depend on the geography of the area and the nature of crimes which are occurring in the area.