I dont mean to sound rude to those people that work for Eastern Airways, but as a former cabin-crew member with bmi regional and still using them occassionally for flights, you only have to compare the service and the prices to realise which is better.
People feel more comortable on jets and bmi regional operate an all-jet fleet using ERJ-135 and ERJ-145s, which are equipped with all-leather seats and are reasonably comfortable and quiet.
Onboard passengers recieve free drinks and a free snacks, and on out of EMA we used to serve hot food on breakfast flights.
Also, bmi's new fares that are bringing them inline with low-cost airlines are extremely cheap compared to the very expensive starting prices Eastern are asking on some of the ex-BA routes.
I dont know about Eastern, but they are up against a great airline service wise.