What precisely differentiates Computer Science from Mathematics in theoretical context?
Theoretical computer science is what theoretical computer scientists do; and mathematics is what mathematicians do. Other than that, there is no accepted definition of either. One might argue that theoretical computer science is a particular branch (or branches) of mathematics, influenced (at least originally) by the problem of efficient computation.

Many areas of mathematics are clearly not theoretical computer science, say functional analysis, category theory, algebraic geometry, algebraic number theory, and many others. However, sometimes these areas are applied to theoretical computer science. Parts of them might then form part of theoretical computer science, if there is a community of theoretical computer scientists who decided to work on them.

On the other hand, computability theory is not necessarily part of theoretical computer science, depending on which community of theoretical computer science one takes as reference. Theoretical computer science (or at least its part known as "Theory A") is traditionally about what can be efficiently computed rather than about what can be computed at all.

Many mathematical theorems have no computational content, and in some cases this can be made precise. One example is Kawamura's result that ODEs are hard to solve in general. It is not true that every mathematical proof has constructive content – indeed there are mathematical techniques which are inherently non-constructive, for example the use of compactness or the axiom of choice in any of its other guises. 

If you are interested in the computational content of mathematical statements and proofs, you might want to look into reverse mathematics and bounded arithmetic (sometimes known as feasible mathematics or feasible arithmetic).