There are two related major differences between Locke and Hume, their focus and their conception of science. Locke's is focused on the knowledge new experimental science provides, he is interested in the epistemologically clarifying the status of this knowledge, but he largely takes most of it for granted (e.g. he accepts Newton's corpuscular theory at face value). Locke even paraphrased the medieval "philosophy is the handmaiden of theology" into "philosophy is the handmaiden of the sciences". Hume's focus, on the other hand, is first on the human nature, specifically human reason, itself. It is it that he wishes to subject to the "scientific method" first, and then decide what it is that science can and can not provide, and how what it does provide is to be properly understood.
Locke amends the Aristotelian "scientia", as a collection of necessary truths connected by syllogisms, into something friendlier to the experimental approach, but he still retains it as an ideal attainable say in Euclidean geometry. It is just the limitations of human senses in their penetration into the "real essences" that prevent us from attaining this ideal:
"I doubt not but if we could discover the Figure, Size, Texture, and Motion of the minute Constituent parts of any two Bodies, we should know without Trial several of the Operations one upon another, as we do now the Properties of a Square, or a Triangle.
Had we Senses acute enough to discern the minute particles of Bodies, and the real Constitution on which their sensible Qualities depend, I doubt not but they would produce quite different Ideas in us; and that which is now the yellow Colour of Gold, would disappear, and instead of it we should see an admirable Texture of parts of a certain Size and Figure."
Still, even with the imperfect sensory input our "innate capacity to reason" produces contingent but probable knowledge worthy of the name, and it is there that Locke's natural philosophy breaks with Aristotle.
Hume's prior analysis of human faculties leaves him far more sceptical than Locke about their fruits. He rejects the "innate capacity to reason", and his analysis of our sensory input convinced him that the "unities" which Locke and even Berkeley still saw there, were mere "bundles of perceptions" lumped together by frequent repetition. This included even the sacrosanct dogmas of the inner "unity of self", and the outer "chains of causation", retained by the experimental science from the traditional metaphysics. Hume does single out mathematics as more trustworthy, but that is only because it covers "relations of ideas" only, and no "matters of fact". Contra Locke, it can not therefore serve as the ideal for empirical sciences. The role of the latter reduces to associating experience to form useful habits. That is Hume's "mitigated" scepticism, see Hume: Epistemology on Philosophy Pages.
Thus, while Locke takes methodological, and even some factual, assumptions of the new science for granted, and explores their implications, Hume subjects the very sources of scientific knowledge, human reason and senses, to the same interrogative scrutiny that science gives the world. In that, and in the skeptical conclusions that followed, he anticipated Quine's naturalized epistemology. It does raise the concerns about the circularity of justification of science by science, but Hume is not yet troubled by them as Kant will be after him. Much later Quine will defend it with:"such scruples against circularity have little point once we have stopped dreaming of deducing science from observations". In this, Hume's "psychological" approach to the grounding of science anticipates Peirce's and Quine's, and Quine explicitly credits him as a major inspiration in Epistemology Naturalized:
"It was sad for epistemologists, Hume and others, to have to
acquiesce in the impossibility of strictly deriving the science of
the external world from sensory evidence... The old epistemology aspired to contain, in a sense, natural science; it would construct it somehow from sense data. Epistemology in its new setting, conversely, is contained in natural science, as a chapter of psychology. But the old containment remains valid too, in its way."
