There more like different coins.
Clinton used his executive authority to make laws (an opinion, which I think is accurate).
Bush (and Reagans) signing statements don't nullify the laws, in that they are not stating that they refuse to follow them, they are stating how they will do so. Congress leaves them (unintentionally) the leeway to do so, by not making detailed bills.
An executive order can easily be overturned, by another order, a congressional bill with a signature, or judicial review.
A signing statement doesn't even have that kind of power, and a president that simply uses his signing statement to nullify a bill, has just effectivly vetoed it, if not used it as a line item veto, something, that Bush has NOT done, nor like all previous executives from before Nixon has he used it to embargo funds (he lacks the authority due to a watergate era budget act).
The signing statement is not a line item veto, and can not be used as one, it is not a means to embargo funds can no more be used to do those 2 things then an executive order can repeal the law of gravity.
Bush has however used his signing statement as an advisory opinion, which has the same weight as a friend of the court brief, but no power, unless someone in court, uses the signing statement as proof of a laws unconstitutionality or cites it as a reason to overturn it.
Here's the problem with that argument: who among the three branches of government executes the laws?
The Prez.
So in this case, Bush's statement carries near-absolute power as long as he is president - because he has stated he will NOT execute the provisions in question. Which only leaves the recourse of a court battle.