Both mobile communication and mobile computing involve wireless data transfer. The difference is in the kind of data being transferred, and the kind of service being provided.
The prevailing definition of mobile communication is any kind of communication that is done over a mobile phone interface. People often use the terms "mobile communication" and "wireless communication" somewhat interchangeably. The main idea, though, is that mobile communication implies voice communication, and unless someone who is talking about mobile communication specifies data, many readers or listeners assume they’re talking about voice communications.
By contrast, the term mobile computing specifically involves the kinds of data transfer that we think of as data, not voice. Telecom providers have done a good job of distinguishing between these two types of services, traditionally separating data from voice charges and vice versa. That makes sense, because the inherent structures for sending either voice or data communications are different, even though voice has become much more like data as analog systems are replaced by digital ones.
Another interesting point is that there’s a different time frame for mobile communications compared to mobile computing. Anyone who understands the history of the telecom industry through the last 20 years will be familiar with the fact that individual consumers were using cellphones long before they were able to use them for mobile computing. First, big telecom companies set up dedicated wireless networks used to send voice communications and data packets through wireless networking systems. These were significantly different from the global IP network or Internet we can access today via a smartphone. It’s only recently that telecom companies have been able to provide more and more data through these same systems, which has led to the common use of mobile devices to stream movies, respond to emails, or accomplish other digital tasks, many of which are Internet-based.
It’s also important to note that these modern mobile computer tasks are different from SMS text messaging, which predated them by several years. SMS text messaging is a significantly different service than mobile computing because it only offers the ability to send character-based text messages instead of voice through a particular interface. New mobile computing services, on the other hand, allow for different kinds of data manipulation and two-way digital communication. For a very evident example of the difference between mobile communications and mobile computing, take a "pre-smartphone" cellphone and set it next to a modern iPhone, Android or other smartphone. The difference in layout and interface is a physical example of the difference between a phone that offers mobile communications, and one that offers mobile computing.