differential geometry - Where in fundamental physics are Lie groups actually important (and not just Lie algebras)?

I was wondering where in fundamental physics the global structure of a Lie group actually makes a difference.

Most of the time physicists are sloppy and don't distinguish groups and algebras properly. However, although we talk about groups all the time I wasn't able to come up with an instant where we don't actually care only about the corresponding Lie algebra.

As an example, physicists usually talk about the Poincare group. However, the thing we are really interested in is the corresponding complexified Lie algebra, which actually belongs to the universal covering group

$$ SL(2,\mathbb{C}) \ltimes \mathbb{R}(3,1). $$

Now, the other kind of symmetry that is important in fundamental physics is gauge symmetry. However, again the global structure doesn't seem to be important. To quote from Witten's Physics and Geometry:

“Experiment tells us more directly about the Lie algebra of G than about G itself. When I say that G contains the subgroup SU(3) X SU(2) x U(1), I really mean only that the Lie algebra of G contains that of SU(3) X SU(2) X U(1); there is no claim about the global form of G. For the same reason, in later comments I will not be very precise in distinguishing different groups that have the same Lie algebra.”