In context|computing|lang=en terms the difference between application and client is that application is (computing) a computer program or the set of software that the end user perceives as a single entity as a tool for a well-defined purpose (also called: application program; application software) while client is (computing) the role of a computer application or system that requests and/or consumes the services provided by another having the role of server.
As nouns the difference between application and client is that application is the act of applying or laying on, in a literal sense; as, the application of emollients to a diseased limb while client is a customer, a buyer or receiver of goods or services.