Edge computing has emerged as a popular paradigm for supporting mobile and IoT applications with low latency or high bandwidth needs. The attractiveness of edge computing has been further enhanced due to the recent availability of special-purpose hardware to accelerate specific compute tasks, such as deep learning inference, on edge nodes. In this paper, we experimentally compare the benefits and limitations of using specialized edge systems, built using edge accelerators, to more traditional forms of edge and cloud computing. Our experimental study using edge-based AI workloads shows that today's edge accelerators can provide comparable, and in many cases better, performance, when normalized for power or cost, than traditional edge and cloud servers. They also provide latency and bandwidth benefits for split processing, across and within tiers, when using model compression or model splitting, but require dynamic methods to determine the optimal split across tiers. We find that edge accelerators can support varying degrees of concurrency for multi-tenant inference applications, but lack isolation mechanisms necessary for edge cloud multi-tenant hosting.