...ergo propter hoc

So here's my issue, what is the functional difference between cum hoc ergo propter hoc and post hoc ergo propter hoc?
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crobert: post = after, cum = with
crobert: post implies causation while cum just implies a correlation
Peter: thank you
Peter: no, I meant what is the functional difference
crobert: post needs a strict chronological sequencing
crobert: while in cum they just need to be related
Peter: in what example
Peter: would there not be some kind of chronological sequence
Peter: if causation were implied?
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Naturally, Wikipedia was the first source to be consulted.
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crobert: ok i'll just steal one from wiki
crobert:

"Sleeping with one's shoes on is strongly correlated with waking up with a headache.
Therefore, sleeping with one's shoes on causes headache. "

crobert: here you have "sleeping" and "with shoes on"
crobert: one doesnt happen after another
crobert: in fact, they're going on at the same time
crobert: as opposed to

"i saw a black cat, then crashed my car. therefore seeing black cats cause car crashse"

Peter: there is clearly a chronological sequence
Peter: Sleep with shoes ---> wake up with head ache
Peter: arrow is time

crobert: ok say you had
crobert: number of lung cancer cases has been steadily increasing
crobert: and
crobert: number of cigarettes sold has been steadily increasing
crobert: so you falsely conclude that they are related
crobert: one doesnt necessarily happen after the other
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So basically, I'm not seeing it. If you're implying that a causes b, then D[b(t)] must equal kD[a(t+x)]. Where t = time and {x|x ∈ R, x > 0}. The k could even be a function of a(t) or something, but you can't just say they're both positive and therefore correlate. If your k value ends up looking like a Weierstrass function, your correlation is garbage in the first place without even getting into the causation part.

Any philosophers reading this, feel free to drop me a line.

Unless your name is Kant. In which case: don't.

update:
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Peter: I thought of something
Peter: dead people are old
Peter: therefore time causes death
Peter: I think this is the only way around the chronological factor
crobert: <_<
crobert: you're just thinking it wrong
crobert: it works, you just think it doesnt
crobert: you have two simultaneous events
crobert: the fallacy is that one is dependent on the other
Peter: how can a cause b if they both happen at once
Peter: unless one of them is time
crobert: protip: it doesnt
crobert: that's why it's a fallacy
Peter: this is a shitty fallacy
...
crobert: what part of it dont you get
crobert: you understand that it's a fallacy right <_<
Peter: so cum hoc is the universal set
crobert: yeah i sort of said that already
Peter: but pos hoc' = half diminished
Peter: That's the crux of my issue
Peter: maybe I'm looking at this wrong
Peter: maybe it's stating the obvious
Peter: AKA 2 simultaneous events can't cause each other because it's simultaneous
Peter: AKA this is a shitty fallacy
crobert: all fallacies are shitty
crobert: give me a nonshitty fallacy
Peter: naturalistic fallacy is alright
Peter: I am going to eat my GM foods and enjoy them