What is the difference between a formula and a proof? As I understand it, a formula is a method for solving a mathematical problem expressed using alpha numeric characters like the quadratic formula is a method for solving quadratic equations when factoring will not work. I understand a proof to be a logical argument that may or may not produce a formula, but will produce a statement that something is true or false mathematically.

Take for example: 
$$x^2+13x+22=0$$
This equation will not factor, so we would use:
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ 
to solve for x.

However, does the fact that this formula always allows you to solve for x constitute a proof or "scientific proof" that this formula speaks to mathematical and thus scientific truth? Or, must a formula have a proof in order to really be considered valid scientifically?
The only way that I know a formula works is via a proof. I might be able to verify by hand that it works for a number of cases, but that doesn't mean it always works; see here and here for some examples of this. This is not to say that experimental evidence is worthless - quite the contrary. But the special role of a proof is something which cannot be ignored.

Now, there are subtleties here. In order to prove something, I need to begin with axioms. What axioms are "acceptable?" The standard axiomatic foundation of mathematics is ZFC (but see here), but there are some "concrete" problems which can't be proved using these axioms alone (see e.g. here). The existence of such problems - and Goedel's theorem more generally - shows that ultimately, the notion of "proof" is more nuanced than we might think at first. For example, there could be a formula that always "works" for a given concrete problem, yet can't be proved to always work inside ZFC. 

However, this situation tends to be the exception rather than the norm. And the answer to your question is no - formulas and proofs are quite different!
Man, here is a good example of formulas and proofs being different. Imagine trying to find a formula for a 5th degree polynomial, you keep dialing away at your calculator making up tables of values. You punch and punch away until your fingers a sore looking for patterns, just to find out that someone along time ago by the last name Galois used logic and proof to prove that there is no formula for a 5th degree polynomial!