Weak collision resistance (CR), or second-preimage resistance, is the property that given $x$ and $h(x)$ ($h$ a hash function) it's difficult to find $x' \neq x$ such that $h(x') = h(x)$. Strong CR, or just collision resistance, is the property that it's difficult to find any two $x,x'$ with the same hash value.
The latter is easier than the former; the former is the "hash function analogue" of "same birthday as me" and the latter the analogue of "any two persons sharing a birthday".
Now then, what would a collision attack look like? I don't understand when it would be of relevance that it's relatively easy to find a pair $x,x'$ with the same hash value; isn't an adversary typically confronted with a given hash value that he has to "work with"? I.e., $x$ and $h(x)$ are given, if he can find a collision pair $(y,y')$, well, what does that help him?
Also, how does one actually find "any two" inputs with the same hash value? Seems to me that you must always first choose one input $x$, and then compare against that, at which point we're back to finding second preimages, right?