Richard Mottram, who retired last month from his post as an adviser on intelligence and security, said focusing too tightly on the terrorist movement risked underplaying other important threats such as climate change or a possible flu pandemic, a British newspaper reported.
"What we shouldn't do is play into al Qaeda's hands by exaggerating the extent and nature of the threat they present globally," Mottram was quoted by The Observer newspaper as saying.
"This focus is not smart when it comes to dealing with people who are trying to make us think that they are the greatest threat," he told the paper.
Among the significant problems Mottram identified deserving of equally serious attention: global warming; flu pandemics; the emergence of rogue states; globalization and its impact on power balances; global poverty and its impact on population movement; energy security; the proliferation of weapons of mass destruction; and organized crime.
The former official's message is in contrast to the assessment of Jonathan Evans, head of domestic spy agency MI5, who used a rare public speech last month to warn of a growing terrorist threat.
He said about 2,000 people in Britain are suspected of being involved in terrorism planning and claimed there is evidence extremists are grooming children and teenagers for attacks against Britain.
"Al Qaeda has a clear determination to mount terrorist attacks against the United Kingdom," Evans said in the speech.
Mottram, who as permanent secretary at the Cabinet Office advised Brown and former Prime Minister Tony Blair, said ministers need to understand the scope of all potential threats.
"We need to have a better understanding of the context in which we operate and the breadth of challenges we face," the newspaper quoted him as saying.