The previous answer is great and gives some good examples. Another "reason" for ILP that I might add is that it is often the most obvious and most straightforward way to represent the selection of discrete alternatives (for example a binary variable X_ j is 1 or 0). I'd even ask what would you do instead? Perhaps represent X_ j as a real number bounded by 1 and zero (relaxation of the problem). This is a strategy for attacking the case, but you would have to agree that fractional solutions are not representative of the real problem that we have set. (This occurs in facility location, portfolio selection, and many other areas.) So the problem might require branch and bound or other implicit enumeration schemes to finish off the selection of only integer solutions.

I think that if you want to understand some of the beauty (and difficulty) of ILP check out the knapsack problem. Almost as simple as you could imagine, and yet there are integer variables with real meaning and significance for this model. While this is "difficult" it has not stopped clever algorithmic approaches that improve over the most obvious greedy heuristic.

NP-Complete problems are compute bound the sense that the amount of computation cannot be bounded in a polynomial manner as the number of data elements or restraints grow.  For smaller problems, you can do the computation and many of the ILP methods are exceptionally efficient (meaning that you cannot do the computation faster with alternative algorithms or software).

With some (many?) of the NPC complete situations, it is possible to apply heuristics (ad hoc algorithms with no theoretical guarantees) on parts of the computation that can speed up the computation for certain (much narrower) computation that may apply with certain classes of problems that are particular to specific subject-matter areas.

The following provides an example of where a heuristic helped improved the computational speed of set-covering algorithms applied to statistical data editing by a factor of 100. 

Winkler, W.E. (1997a), “Set-Covering and Editing Discrete Data,” American Statistical Association, Proceedings of the Section on Survey Research Methods, 564-569 (also available http://www.census.gov/srd/papers/pdf/rr9801.pdf).

The above algorithm/software when combined with additional algorithms providing a 100-1000-fold increase in computational speed can yield generalized software for the clean-up and preliminary analysis of sets of national files.

Winkler, W.E. (2013c), Cleanup and Analysis of Sets of National Files, Federal Committee on Statistical Methodology, Proceedings of the Bi-Annual Research Conference, http://www.copafs.org/UserFiles/file/fcsm/J1_Winkler_2013FCSM.pdf.,  https://fcsm.sites.usa.gov/files/2014/05/J1_Winkler_2013FCSM.pdf.

With the new methods, a skilled group of individuals can do the work in 3-6 months; with methods/software that are 100-1000 times as slow (as with the fastest conventional software), it is not clear how long the clean-up would take.

The previous answer is great and gives some good examples. Another "reason" for ILP that I might add is that it is often the most obvious and most straightforward way to represent the selection of discrete alternatives (for example a binary variable X_ j is 1 or 0). I'd even ask what would you do instead? Perhaps represent X_ j as a real number bounded by 1 and zero (relaxation of the problem). This is a strategy for attacking the case, but you would have to agree that fractional solutions are not representative of the real problem that we have set. (This occurs in facility location, portfolio selection, and many other areas.) So the problem might require branch and bound or other implicit enumeration schemes to finish off the selection of only integer solutions.

I think that if you want to understand some of the beauty (and difficulty) of ILP check out the knapsack problem. Almost as simple as you could imagine, and yet there are integer variables with real meaning and significance for this model. While this is "difficult" it has not stopped clever algorithmic approaches that improve over the most obvious greedy heuristic.

I agree with the notion from Dr. O'Kelly - fractional solutions are not representative of the real problem. We developed a discussion paper over a decade that describes why one might want to develop spatial forest plans (which may require integer decision variables). Perhaps this can inform the "why" question...Journal of Forestry (2003) 101(2): 24-29.