Many people think that if you use a washer with your application, you are using a grommet. This is not the case. The main difference between an eyelet and a grommet is that the grommet has a larger flange diameter in relation to the hole size than an eyelet. The larger flange is used for additional holding strength in high-stress applications such as tents, banners, and flags. For most applications, an eyelet with a washer will provide more than adequate holding power, if set properly.
Grommets tend to look more "industrial" and do not have the "finished look" of an eyelet. Eyelets are the main choice in garments, belts, hats, shoes, and other applications where aesthetics matter. People also often choose to use eyelets when binding paper, booklets, calendars, and for hole reinforcement in tags and labels. The small flange of the eyelet does not detract from the overall appearance of these items since it is not as pronounced as the flange on a grommet.
Below is a picture to illustrate the point. On the left is an eyelet with the same inside hole diameter as the grommet on the right. Notice how much larger the flange on the grommet on the right is versus the eyelet on the left.