Since both have multiple definitions, it's hard to say.

Proof Justifies belief
Proof denies and destroys faith

From dictionary.com:

Faith

noun

1. confidence or trust in a person or thing: faith in another's ability.
2. belief that is not based on proof: He had faith that the hypothesis would be substantiated by fact.
3. belief in God or in the doctrines or teachings of religion: the firm faith of the Pilgrims.

Belief

noun

1. something believed; an opinion or conviction: a belief that the earth is flat.
2. confidence in the truth or existence of something not immediately susceptible to rigorous proof: a statement unworthy of belief.
3. confidence; faith; trust: a child's belief in his parents.

Depends on definition and usage. Properly, in my view, a belief is formed around a perception (rightly or not) that the preponderance of evidence (valid or not) leans significantly in a particular direction. In other words it is at least nominally evidence-based.

Religious faith is affording belief without a requirement of substantiation or the deployment of any sort of skeptical or critical thinking or objective observation. In general it grants belief to asserted truth rather than actual reality.

Colloqual faith, is a near polar opposite to religious faith. It is actually just trust or at times hope, based on experience or sound reasoning. It has as its input, some fact(s) of existence that would lead a reasonable person to consider this trust or hope to represent a reasonable expectation. Whereas religious faith would just declare by fiat that a thing is so, with no more basis than an assertion of religious dogma (from scripture or from theological authority-figures).

Online dictionary definition of faith:
"confidence or trust in a person or thing:
faith in another's ability.
belief that is not based on proof:
He had faith that the hypothesis would be substantiated by fact.
belief in God or in the doctrines or teachings of religion:"

Online dictionary definition of belief:
"something believed; an opinion or conviction:
a belief that the earth is flat.
confidence in the truth or existence of something not immediately susceptible to rigorous proof:
a statement unworthy of belief.
confidence; faith; trust:"

one probably works whilst the other is a bit of a lottery

Well, belief is mental, and faith is fanatical. You have beliefs in certain things because of what you've observed, while faith is experienced without evidence.

Let's put it like this:

"I believe I left the oven on." That makes sense as a sentence.

"I have faith that I left the oven on." That makes no sense.

It's easier to illustrate the difference than to define it. A belief is usually based on some kind of rational evidence. Faith is based on no evidence whatsoever.