Believe it or not, there is not much difference between the two. In fact, they are the same, but with different names. Broderie Anglaise is the French name for "English Embroidery," which just makes it sound a touch more fancy.  Its main characteristics are the oval and round patterns pierced on the fabric. You could say that eyelets are the pattern present on English Embroidery, but it is common to see people just calling them eyelet fabric.