The use of any as opposed to every.

One problem with "any" is that, in mathematical English, it can mean either "there exists" or "for all".

Some examples where "any" means "there exists":

• A number $\lambda$ is an eigenvalue of a matrix $A$ if any nonzero vector $x$ satisfies $(A-\lambda)x = 0$.

• $X \cap Y$ is nonempty if any element of $X$ is an element of $Y$

Some examples where "any" means "for all":

• A set $A$ generates a group $G$ if any element $g \in G$ can be written as a product of elements of $A$.

• A graph is connected if any two vertices are connected by a path.

The result of this is that you have to use context or prior knowledge to read a mathematical sentence with "any" in it; there is not a general algorithm that will tell you how to read English. I generally advise my own students to avoid the word "any" until they are very comfortable with mathematical English.

In the context of the question, if you state the converse as "If every pair of vertices of a graph is connected by a unique walk, then the graph is a tree", you can avoid the "any" issue altogether.

Logically, they're the same: "For any $x$, $P(x)$" is the same as "For every $x$, $P(x)$" is the same as "$\forall x$, $P(x)$." Your initial inclination is correct.

There is, however, at least to my ear, a difference in connotation and hint of proof method. "For any" emphasizes that this property holds for an arbitrary element --- in your case, an arbitrary pair of vertices on the tree. Perhaps the proof will include the choice and consideration of an arbitrary object.

Meanwhile, "For every" emphasizes somehow some sort of "uniformity" --- perhaps the proposition can be proven from the properties of this class of object without choice of an arbitrary object.

But a note of caution: These are subtle connotations, certainly not definitions set in stone! Don't take them as gospel, but rather merely as one person's coloring of the language. They are two ways of phrasing the same logical quantifier.