The difference is in degree. To illustrate, imagine the cross section of a non-bevelled, non-chamfered part. It will have 4 surfaces, two in each plane- vertical and horizontal- and each pair parallel. There will be four 90-degree corners.
A bevelled edge will still have 4 surfaces, but two of them are no longer parallel and one corner comes to a sharp point- often at 45 degrees, but not by definition.
A chamfered edge will have 5 sides, because the angled surface does not meet the lower horizontal. There are still two pairs of parallel sides with the addition of one angled surface, typically at 45 degrees but again, not by definition.
A good example is a kitchen countertop. Kitchen countertops are often referred to as "bevelled" when they actually are chamfered. This is to avoid confusion- the general public don't know what a chamfer is.