In context|computing|lang=en terms the difference between service and client is that service is (computing) a function that is provided by one program or machine for another 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.
In context|legal|lang=en terms the difference between service and client is that service is (legal) the serving, or delivery, of a summons or writ while client is (legal) a person who employs or retains an attorney to represent him or her in any legal matter, or one who merely divulges confidential matters to an attorney while pursuing professional assistance without subsequently retaining the attorney.
As nouns the difference between service and client is that service is an event in which an entity takes the responsibility that something desirable happens on the behalf of another entity or service can be service tree while client is a customer, a buyer or receiver of goods or services.
As a verb service is to serve.