Which version of the Pigeonhole principle is correct? One is far stronger than the other One version of the Pigeonhole principle says that if the cardinality of a set $A$ is greater than that of a set $B$, then there can be no one-to-one function that maps from $A$ to $B$. 
Another version says: If $n$ elements are partitioned into $m$ subsets, then at least one subset must contain at least $\lceil n/m \rceil$ elements. Note that the $\lceil \ \rceil$ enclosing the $n/m$ tells us to round the value up.
The latter version is just a more powerful version, no? It seems strange to me that one version of a theorem can objectively give more information than another, so it's likely that either the latter version is not correct or I am incorrect in thinking that it somehow has more utility.
Or is this normal? Is it common for certain versions of theorems to simply have less utility than other versions?

Which version is more useful depends on the situation. There are many situations where all that's needed is one hole with more than one pigeon, and besides that, the number of pigeons in each hole is not important. 
As an additional note, for infinite sets, the nature of the principles changes. The definition of cardinality is equivalence classes defined in terms of injectivity, so saying that if |A| > |B|  then there's injective function is pretty much just a restatement of the definition. And for infinite sets, if |A| > |B|, then, to the extent that |A|/|B| is defined, it is equal to |A|. For instance, if you assign to each integer a set of real numbers, and each real number is in at least one such set, then at least one set has the same cardinality as the set of real numbers.
