TITLE: Difference between $\implies$ and $\;\therefore\;\;$?
QUESTION [26 upvotes]: I've seen both symbols used to mean "therefore" or logical implication. It seems like $\therefore$ is more frequently used when reaching the conclusion of an argument, while $\implies$ is for intermediate claims that imply each other. Is there any agreed upon way of using these symbols, or are they more or less interchangeable?

REPLY [7 votes]: $$A \implies B \implies C \tag{1}$$ has a different meaning and reading from $$A;\:\:\therefore B;\:\:\therefore C \tag{2} \\\text{($A$ is true; therefore $C$ is true)}.$$ Line $(2)$ asserts premise A, that $A \implies B$ and that $B \implies C,$ and concludes that $C$ is true.
Line $(1),$ on the other hand, neither specifies that sentences $A$ and $B$ are actually true, nor asserts that sentence $C$ is true. It in fact has three possible meanings, no pair of which is equivalent (convince yourself using the truth assignments $(0,0,0)$ and $(1,0,0)$): $$A \implies (B \implies C).\tag{1a}$$
$$(A \implies B) \implies C\tag{1b}$$
$$(A \implies B)\: \text{ and } \:(B \implies C)\\\text{(A being true implies that C is true)}\tag{1c}$$ The third is the most common interpretation, and can be read as “in case $A$ is true, then $C$ too is true”.
If the difference between lines $(2)$ and $(1\text c)$ seems trivial and just a matter of contextually inferring the stronger meaning as necessary, then contrast this true statement \begin{align}\lvert2x\rvert=x-1&\implies\pm2x=x-1\\\text{i.e., }\:\lvert2x\rvert=x-1&\implies x=-1 \;\text{ or }\; \frac13\end{align} with this patently false assertion $$\lvert2x\rvert=x-1; \;\text{therefore, }\; x=-1 \;\text{ or }\; \frac13.$$ (The anomaly is explained here.)
All in all, I think it is bad practice to treat $\implies$ and $\,\therefore\,$ as interchangeable.
Here are several articles expounding their distinction:

implication versus conclusion
“implies” versus
“therefore”
guide to writing mathematics (p. 17)
abuse of the implication
symbol