Wars on abstracts usually solve short term problems but create long term problems. This is because declaring the war solidifies support for those that are opposing the abstract (terror, poverty, drugs, etc.) but in the long run it is nigh impossible to ‘win’ a war on an abstract and if you don’t win a war... ...well the other option isn’t ever politically convenient.
So if President Bush hadn’t declared a war on terror it would in the short run have maybe not galvanised international support. At the time it was hard to determine whether this galvanisation was necessary, e.g. compare the use of the ‘with us or, against us’ trope that President Bush also employed for the same reasons. Probably in reality nothing changes short term.
Long term it means there is one less rhetorical issue as a millstone around Bush and Obama (unless you think this non-use of the phras will butterfly Obama’s presidency). The ‘war on terror’ is a nuisance but at most it’s just colouration on US foreign policy.