Both the branches are good and demanding as well but there are few differences which you can understand below -

EEE: In EEE, students have to study Electrical as well as Electronics. It is the study and application of electrical systems for use in different environments and contains basic electronic courses and more power system.

ECE: Here, aspirants deal with Electronics with some part of communication. The subject uses the scientific knowledge of the behaviour and effects of electrons to develop devices, systems, or equipment that uses electricity as part of its driving force. Also it contains basics on electrical machines and more on integrated circuits and communication systems.

Placemen wise :
EEE: In EEE the competition is not very tough and the availability of jobs too are good. Electrical graduates are employed in Electricity Boards/Utility companies responsible for installation, maintenance, operation of power handling equipment and systems. 

ECE: In ECE, the scope to land into Govt jobs is great. An electronics engineer can find a job in Central Government, State Governments and their sponsored corporations, private industries dealing in manufacture, sales and services of electronics consumer goods and appliances.

Hope this helps!!!