Do you know of any problems (preferably at least somewhat well known), where, for a practical problem size, an exponential algorithm runs much faster than a best-known polynomial time counterpart.
For example, suppose a problem has a practical size* of $n = 100$ and there are two known algorithms: One is $2^n$ and the other is $n^c$ for some constant $c$. Clearly for any $c > 15$, the exponential algorithm is preferred.
*I guess practical size would mean something commonly found in the real world. Like the number of trains on a network.