In day to day, practical programming terms, the biggest difference is in the fact that with SOAP you are working with static and strongly defined data exchange formats where as with REST and JSON data exchange formatting is very loose by comparison. For example with SOAP you can validate that exchanged data matches an XSD schema. The XSD therefore serves as a 'contract' on how the client and the server are to understand how the data being exchanged must be structured.
JSON data is typically not passed around according to a strongly defined format (unless you're using a framework that supports it .. e.g. http://msdn.microsoft.com/en-us/library/jj870778.aspx or implementing json-schema).
In-fact, some (many/most) would argue that the "dynamic" secret sauce of JSON goes against the philosophy/culture of constraining it by data contracts (Should JSON RESTful web services use data contract)
People used to working in dynamic loosely typed languages tend to feel more comfortable with the looseness of JSON while developers from strongly typed languages prefer XML.