How does this apply to MATLAB? This really is a question purely about linear algebra.
Are they the same? No, they are not. Yes, they look somewhat alike in what they return, ALMOST.
The smithForm applies ONLY to square matrices. But also, it applies ONLY to integer valued matrices. READ THE HELP!
Next, try it out. Don't know if they are different? TRY IT.
[U2,S2,V2] = smithForm(A)
They don't look the same to me. But then I do need a new set of glasses.
Finally, if you try using smithform even on a square matrix that has non-integer elements, it will fail.
The Smith normal form is designed to solve a totally different set of problems compared to when one would use the svd. They are different tools, with different properties, different purposes, and different results.