I think of a heuristic more as a procedure that works only in certain cases, than as a numerical approximation. A good example is solving differential equations. There's no algorithm for that AFAIK -- all we have is a mental database of forms of such equations that have been solved. When faced with one to solve, we pattern-match it against our database, and if we have a match, we know the solution. This works okay as long as our database is big enough to contain all the kinds of equations we actually care about for whatever domain we're modeling, but it's always possible to come up with one whose solution isn't known yet.
(Someone will correct me if I have the facts wrong, but even if this isn't true for differential equations, there are certainly areas of mathematics where it is true.)
That is certainly one definition and use case of heuristics.
Another example might be the travelling salesman problem, where we know how to calculate the exact solution for every instance, but it is prohibitively expensive (unless P=NP). There are a number of heuristics that "can find solutions for extremely large problems (millions of cities) within a reasonable time which are with a high probability just 2–3% away from the optimal solution" (Source: https://en.wikipedia.org/wiki/Travelling_salesman_problem)
In that case, you can always come up with a solution, it's just not always the best solution.
(Someone will correct me if I have the facts wrong, but even if this isn't true for differential equations, there are certainly areas of mathematics where it is true.)