There is no difference between the two handsets.
I assume the D600e was on Orange. Orange usually add an e to the phones they approve for use on their netowrk.
I am under the impression it stands for enhanced, as they have to enhance their phones to use some of the services they offer on their network, not all phones support line 2 for example.
Edge -
EDGE (Enhanced Data rates for GSM Evolution) provides data transfer rates significantly faster than GPRS or HSCSD. EDGE increases the speed of each timeslot to 48 kbps and allows the use of up to 8 timeslots, giving a maximum data transfer rate of 384 kbps. In places where an EDGE network is not available, GPRS will automatically be used instead. EDGE offers the best that can be achieved with a 2.5G network, and will eventually be replaced by 3G.