In common usage, fact and truth are often to express a strong degree of synonymy. In the philosophies of mathematics and science, things become a little more complicated because they are closely related, but not necessarily the same thing. For example, some thinkers might argue there are facts, but that they do not express truths, such as the scientific instrumentalists. As such, the philosophies of mathematics and the sciences host a variety of opinions on the relationship between facts and truth, depending on things like which theory of truth is considered.
From MW's entry for fact:
Essential Meaning of fact
1 : something that truly exists or happens : something that has actual existence Rapid electronic communication is now a fact.
2 : a true piece of information The book is filled with interesting facts and figures. Those are the (cold) hard facts of the case.
In philosophy, as always, things get more complicated. In WP, the article on Fact:
In philosophy, the concept fact is considered in epistemology and ontology. Questions of objectivity and truth are closely associated with questions of fact. A "fact" can be defined as something that is the case—that is, a state of affairs.10
Facts may be understood as information that makes a true sentence true.8 Facts may also be understood as those things to which a true sentence refers. The statement "Jupiter is the largest planet in the solar system" is about the fact Jupiter is the largest planet in the solar system.9
Fact and truth are tremendously contentious notions in philosophy, period. Facts can be viewed in light of the objective/subjective dichotomy or one might prescribe to a more complicated theory such as intersubjectivity of which there are flavors, such as Daniel Dennett's heterophenomenology. Often one sees philosophers referring to true and false facts to muddy the distinction. Is the current King of France bald? It seems like a factual matter, because it is not an opinion. In a room full of scientists, there's likely to be agreement that he is or isn't. But the question presumes someone who doesn't exist. Does this mean it's not a fact, or that it's somehow nonsense? If it's nonsensical, how come it seems so meaningful? Questions like these drive debates about what truths and facts are, exactly.
Also, truth has several prominent theories such as those based on correspondence, coherence, and deflation. So, when these different epistemic theories are part of different epistemic models, one starts to get variations on what is the relationship between fact and truth. Is it a fact that a wavefunction describes a quantum state of a physical system? Post QM, Sure. But is that same claim a truth? Does a probability function correspond to an actual wave in the universe? That would seem to suggest that mathematical objects compose physical objects. That might not work for eliminative materialists who think probability waves are products of the mind and the mind is just an illusion. You get the idea.
Fact and Truth in Science
Let's take two general perspectives in science, the debate between scientific realists and instrumentalists. A realist believes that objects exist independent of observers, and that they have definitive properties. One that might be considered is locality; objects need to touch to interact. But then what does one make of non-locality which is also known as action at a distance? An instrumentalist can avoid any worries that a realist cannot by simply proclaiming that action-at-a-distance is not a property that inheres to an object so much as a concept we use as a tool to describe what we observe. It is a fact about entangled particles, but not a truth. This is a tough sell to realists. Einstein's famous quotation about God playing dice with the universe illustrates the reluctance to disentangle fact from truth.
Fact and Truth in Mathematics
A same sort of metaphysical debate occurs among nominalist, conceptualist, and realist camps in mathematics. Mathematicians since Pythagoras and Plato have wrestled with the same sorts of challenges. J.R. Lucas in his The Conceptual Roots of Mathematics shows in his first few chapters how philosophical thinking about mathematics seeks some sort of justification for mathematics. He covers in some detail how Euclid's parallel postulate can be stated equivalently or discarded in favor of others, a historical occurrence that surrounded the development of non-Euclidean geometries. This mathematical development led to changes in how physicists, who are often radical materialists, view the universe, which was transformed from one based on Euclidean geometry and Newtonian mechanics to one that embraced non-Euclidean geometry and was enriched by Einstein's relativistic mechanics.
Is a geometry and its first principles, be they truths or postulates, a property of the universe? Do those axioms constitute facts about that universe? Or are they vacuous symbols that are factual because they do not contradict each other and seem logically independent, but make no actual claim about material? Plato was more a rhetorician than a mathematician and invented the deductive argument to try to create a certain path to truth, but even he was aware of it's limits. Valid deductions can be facts, but that doesn't make them truths.
There's a lot to the question you asked, and many working mathematicians and scientists are ignorant of their own philosophical presumption. Professional mathematicians are overwhelmingly Platonic in their beliefs about facts and truth, but it is a fact that their view of math isn't the only view about what constitutes mathematical epistemology. Science often follows a similar trajectory with a wildly divergent set of views that emerged under debate from the demarcation problem of science. Therefore, a sophisticated thinker in science and mathematics makes use of the fact that there are a wide variety of positions that are available about the relationship between fact and truth, and understands the strengths and weaknesses of each position, coming to their own mind about the matter.