As tobyink pointed out, there's no difference in what ends up in $self, but there is a big difference in what's left in @_. If your subroutine isn't expecting arguments, then you can use either method without concern. If you are going to be taking arguments, then you have a choice to make.
Personally, I'm a big fan of using my $self = shift;. For me, it makes the code cleaner and more mentally straightforward. When using shift, the contents of @_ end up being exactly the same as the arguments I passed to the subroutine. If I use assignment from @_, then I have to constantly account for that $self at the first position, complicating things.