Matlab’s norm applied to a matrix gives the induced matrix 2-norm, equal to the matrices’ largest singular value.
Julia’s norm applied to a matrix gives the Frobenius norm, equal to the root sum of squares of the matrix elements.
Presumably Julia uses the Frobenius norm because it’s way cheaper to compute root sum of squares than an SVD.
And also, the title of the OP is wrong. This is a different of matrix norms, not vector norms.