Speech processing is the science concerned with how speech communication works: how speech is produced by the speaker and understood by the listener. It is also concerned with how these processes can be analysed and modelled, and with how these models can be used to develop technologies that also produce and understand speech (synthetic voices, speech recognisers). The science and technology involved are fundamental to the understanding and remediation of disordered speech. The science of speech is thus at the intersection of many disciplines, particularly linguistics, psychology, acoustics, and engineering.
Language processing, in parallel, deals with computational theories of grammar and meaning, and provides access to fundamentals of linguistics as a science and as an engineering discipline. As a science, it is concerned with the fact that language is used as a medium for thought as well as for communication. As an engineering discipline, it is concerned with tools that work: predictive text in telephones, automated personal assistants, web search, and so on. The fact that you are reading this sentence entails that you have taken advantage of one or more language technologies; it would not be visible to you otherwise. The fact that you understand this sentence (or any sentence) begs all of the questions of cognitive science such as what "meanings" are and how people reason with them.