Language is both an instrument used in human thought/reasoning, and an instrument used in communication/social interaction. Let’s call the former the ratiocinative role of language, and the latter the communicative role of language. In their respective discussions of language, Hobbes and Locke affirm both roles for language, but they differ in terms of how they prioritize those roles when introducing their discussions of language.
For example, Hobbes’s discussion in the first part of De Corpore starts by presenting the role of language in aiding memory and ratiocination on an individual level, and then observes that, unless one wishes their scientific discoveries to perish with them, it is useful to have some way of communicating their knowledge to others. Hobbes takes the ratiocinative role as primary, and later acknowledges a communicative role (seemingly in service of ratiocinative goals).
Contrast this with Locke’s discussion at the outset of book three of the Essay (p. 402):
§1. GOD having designed Man for a sociable Creature, made him not only with an inclination, and under a necessity to have fellowship with those of his own kind; but furnished him also with Language, which was to be the great Instrument, and common Tye of Society.[…]
§2. Besides articulate Sounds therefore, it was farther necessary, that he should be able to use these Sounds, as Signs of internal Conceptions; and to make them stand as marks for the Ideas within his own Mind, whereby they might be made known to others, and the Thoughts of Men’s Minds be conveyed from one to another.
Locke goes on to discuss, in later sections, the role of language in our individual reasoning, but it is clear that the communicative role of language takes priority, in his thinking.
I have a suspicion that this difference in orientation about the priority of the communicative and ratiocinative roles of language underwrites some of the differences in their respective theories of language, but at this juncture, I don’t have any specific examples to point to.