First off, lets not discuss SAM's like the S300 and Akash. They are not in the same league as a full fledged BMD.
The Differences that I know of
1) Indian BMD system is a terminal phase system versus the Chinese system which has been tested only in midcourse phase. The Chinese system is pretty much the same system as the ASAT one. Low velocity target at very high altitude with a Kinetic Kill vehicle. The Indian low altitude system has to overcome challenges like speed of the incoming missile being several times higher in terminal phase than in midcourse phase (where the missile is practically coasting). On the other hand the Chinese system has to overcome the challenge of range. For example if the missile is launched from across the Pacific, midcourse would still be several thousand kilometers away from a land based BMD system located on the mainland. So either the interceptor should have range of thousands of kilometers or the system should be based on a naval platform for it to even work. Even so, there will have to be multiple platforms for a fail safe interception.
2) Indian systems (both PAD/AAD and AD1 and AD2) are based on two tiered designs. The overall accuracy of such a system would be the weighted accuracy of both systems. Therefore a 99% accuracy on each tier translates to a 99.99% overall system accuracy. Since the Chinese have had only one BMD test in the terminal phase so far, not much is known about there being a multi tiered system.
3) Kill vehicles : Though India tested a Kinetic Kill vehicle during the last AAD test, just one of the six tests so far have been KKV, the others being proximity detonation. The lone Chinese test has been on a Kinetic Kill Vehicle. However again, it is easier to smash into a slow moving object than a much faster moving one (midcourse versus terminal)
Unknowns : Radars. Though we know what radars India uses or proposes to use, we have no idea what the Chinese are using to detect launches. Ultimately it would depend on who the BMD targets. If its India specific, radars similar to the LRTR would make sense, if it is US specific, then satellites come into the picture. Hope some Chinese members can elaborate.
Overall I think both nations are following different paths to a BMD. India's initial systems were designed to combat the Pakistan specific threat while China has possibly designed their system US specific. The challenge for the Chinese would be longer deployment timelines due to the need for naval platforms. Also if a multitiered system is developed to catch a missile in terminal phase after it has escaped the midcourse BMD, the Chinese will have to develop technologies that can intercept missiles at much higher speeds (a trident warhead will be at Mach 10+ speeds when it enters the mesosphere and Mach 20+ when its altitude is around 30 km)
India on the other hand has to evaluate whether developing a system that intercepts midcourse is worth doing. The advantages of such a system are obvious, the debris fall far away, probably in enemy territory or ocean etc, Higher reaction times if initial interception fails to kick in other systems. The disadvantage would be the need of multiple naval platforms with significant costs.