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Early on, the two terms were bandied about as vendors added features and tried to find the best way to define their products. Over time, UC has come to mean that at least some of the features include real time communications like phone calls. While UC has store and forward components, like email and sending voice mail messages to a unified inbox for later retrieval, UM has only store and forward messaging components.
The argument can still get sticky. Are SMS text messages to your phone real time, or not? Those generally fall under UM. Again, define your communications needs carefully before shopping for a UC, or UM, system. Many experts consider the UC umbrella to include UM.