Most descriptions of Beech start with "Beech is very like Hornbeam other than ...." and the same goes in reverse for Hornbeam. They are quite difficult to tell apart and are often used in the same type of planting eg formal hedging, but for those keen to know the differences, once you know they are quite easily identified.
In terms of the planting situation, beech prefers drier soils whereas Hornbeam is good in moist soils (but not waterlogged). Hornbeam has a quicker growth rate. The leaves of both are very similar but the differences are that Hornbeam leaves have deeper veins and are less “fluttery” than Beech leaves. In autumn, whilst they both turn coppery coloured, and both hold most of their leaves on the tree/hedge throughout the winter before being pushed off by the new foliage in spring, the colour of Hornbeam leaves (shown on the left in the photo) is browner and Beech leaves (on the right) are more copper coloured (not to be confused with copper beech which is the common name for Fagus sylvatica Purpurea - or purple beech being it's other common name). Maybe that makes it easier to remember (copper = beech) but I hope I havent confused by introducing mention of purple beech!
Hornbeam has adpressed buds which are held close to the branch whereas beech has non-adpressed buds which point outwards. This is a useful way of identifying the species in early spring when the buds are pushing off the previous year's foliage which limits the potential to identify the species via the winter foliage colour (which I think is the easiest way!).