The definition of Kruskal rank : maximum value of $k$ such that any $k$ columns of a matrix $\textbf{A}$ are linearly independent, then $k$ is the Kruskal rank of matrix $\textbf{A}$.
How is it different from rank of the matrix $\textbf{A}$ ?
Can you give an example where Kruskal rank and rank of a matrix are different.
The difference is that for the rank it is "there exist $k$ columns" not "any $k$ columns."
A matrix that contains an all zero-columns has Kruskal rank $0$. A matrix that contains the same column twice has Kruskal rank $1$ (or less).
Clearly such matrices can have a higher rank.