I think differential topology has dozens of these results. Here are some examples that immediately come to mind:

• The tubular neighborhood theorem: every submanifold $N$ of a manifold $M$ has an open neighborhood which is diffeomorphic to the total space of the normal bundle of $N$
• The fundamental theorem of Morse theory: if $f: M \to \mathbb{R}$ is a Morse function and $[a,b]$ is an interval which contains no critical values of $f$ then the set of all points where $f \leq a$ is a deformation retract of the set of all points where $f \leq b$
• Every continuous isomorphism of smooth vector bundles is homotopic to a smooth isomorphism (and other such "continuous equivalence = smooth equivalence" results)

Probably most topologists know the basic ideas behind the proofs of these results, but I think many would be hard-pressed to actually write down a complete argument. I say this with confidence because I know of several textbooks by good authors that have proofs which are either wrong or sketchy on some details.

There are also some results with standard proofs that are widely known, but I think considerably more people use the results than know the proofs:

• De Rham's theorem: the De Rham cohomology groups of a manifold are isomorphic to the singular cohomology groups with real coefficients
• The Hodge theorem: every De Rham cohomology class on a Riemannian manifold has a harmonic representative
• Whitehead's result that every smooth manifold has a unique PL structure