Excellent question and is one that certainly comes up time and time again. Both can provide gain stages in your signal chain, however, an op-amp can only be used in single-ended measurements when a high input impedance is required. An instrumentation amplifier can gain up differential signals while rejecting common-mode signals. Additionally, in-amps have an extra reference pin that can be used to level shift the output to set the biasing for any following stages and allows for maximizing output signal swing for various sensor types.
For more information, take a look at A Designer's Guide to Instrumentation Amplifiers found in the education library on analog.com (here), this is an excellent resource, it demystifies the in-amp, where is goes into detail on the fundamentals of in-amps, what they are, how they operate and how and where to use them. Also discussed are some examples of where in-amps can be applied.
