There are three important types of matrix norms. For some matrix $A$
Induced norm, which measures what is the maximum of $\|Ax\|/\|x\|$ for any $x$.
Element-wise norm, which is like unwrapping $A$ into a long vector, then measuring its vector norm.
Schatten norm, which measures the vector norm of the singular values of $A$.
So, to answer your question:
So if by "2-norm" you mean element-wise or Schatten norm, then they are identical to Frobenius norm. If you mean induced 2-norm, you get spectral 2-norm, which is $\ge$ Frobenius norm.
As far as I can tell, if you don't clarify which type you're talking about, induced norm is usually implied. For example, in matlab, norm(A,2) gives you induced 2-norm, which they simply call the 2-norm. So in that sense, the answer to your question is that the (induced) matrix 2-norm is $\ge$ than Frobenius norm, and the two are only equal when all of the matrix's eigenvalues have equal magnitude.