Instead of the pads of the fingers being against the rim, the side of the middle finger is against the rim. This grip promotes a palm-up follow through, and helps stop people from turning their forehands over on release. The down side of this grip is that the snap puts lateral pressure on the finger joints, and persistent hard throws using this grip can damage the joints. It is therefore only recommended as a teaching aid, and not for use by experienced players. Endless variety is possible, particularly with fine-tuning.
Most people use one of the above basic grips, but with their own minor adjustments.
With regard to radical differences, few are of practical use. The only alternate grip to those above which is sometimes seen is the use of one, three or even four fingers inside the rim instead of two. Fewer fingers for some reason seems to give better control and accuracy, but using only the index finger puts a lot of pressure on just the one finger, and this is probably why it is not commonly used. These are essentially the same as for the forehand. The significant difference is that the hammer throw is released upside-down, so that support underneath the disc is not as important as for the forehand.