What impact has the Muslim resurgence had on world history? Why has the West been unable to cope with this development in a realistic manner?

It's important to remember that Islam has been a major
world religion for quite some time, so the term resurgence does not really refer to
membership, or even the mainstream beliefs of the vast majority of Muslims.  The term
generally refers to the resurgence of radical Islam, especially along Wahhabi lines such
as al-Qaeda.

The impact on world history is most obvious in
the military conflicts in the Near East, most notably in Afghanistan, and to a lesser
degree in Iraq.  Radical Islam provoked the invasion of Afghanistan after the 9/11
attacks, the pursuit of bin Laden and al-Qaeda around the globe, and that conflict
widened to include attacks on US forces in Iraq as well.  The impact then, in these
cases, was to force or provoke Western society into investing massive military and
financial resources into responding directly to some elements of the Muslim
Resurgence.

The West has been largely unable to cope with
this radicalization because, and this is a matter of opinion, it has fundamentally
misunderstood or even ignored the root causes of the Resurgence.  It has essentially
declared war on a set of ideas.  Terrorism has no military solution, and probably no
diplomatic one.  The roots and recruiting grounds of radical Islam are economic, and
depend on crushing poverty to be successful.  This is both inconvenient and complicated,
with no short term solution, something the West, and the US in particular, does not
typically handle well in recent years.