From my understanding, a Motion Energy Image just locates where motion happend and not how. A Motion History Image can express how a motion, i.e. the direction of the motion, happend.

See the following links:

http://www.ece.iastate.edu/~alexs/classes/2007_Spring_575X/slides/09_MotionHistory.pdf

page 2 and 3

http://www.ijcea.com/tag/motion-energy-image-mei/

page 5 : "We use only one temporal template that is MEI for extract action features, that is enough to recognize actions, because we don‟t need to use another temporal template; MHI (motion history image), which is decide how action is happened. [...]"

Eventhough you asked this question 2 years ago and probably already got an answer for it, perhaps other people will see this when searching on the internet. I've found it to be not clearly defined anywhere so I thought I'd just post it here.

The motion energy image (MEI) encodes the whole motion into a single binary template while the motion history image (MHI) indicates the recency of the motion using different intensity levels.

The motion history image (MHI) is a view-based temporal template where the temporal motion information is collapsed into a single image template where intensity is a function of recency of motion. In MHI,... It represents motion sequence and direction in a compact manner.

Motion energy image (MEI) is used to characterize the motion regions and does not hive any information about the direction