Advantages of Travelling wave IMS and Drift tube IMS
In a nutshell, since other sources probably would give much more accurate and complete answers:

Advantages of DT-IMS over TWIMS:


Direct access to absolute collision cross sections. This is a major advantage considering the difficulty to provide accurate CCS calibration scales for TWIMS measurement as can be seen in the recent litterature.
Separative resolution is usually reported as higher, although recent work by Dick Smith's group at PNNL seems to show that it is mostly related to separation device lengths which are different.


Advantages of TWIMS over DT-IMS:


Reduced ion losses and increased sensitivity. The stacked-ion ring basis leads to reduced drift towards the sides. This advantage has been reduced by the addition in several DT-IMS setups of ion funnel electrodes which focus the ion trajectories upon reaching the end of the drift tube.
No need for large potentials and thus large voltage generators and all the related problems. To reach a given electric field, the potential difference does not need to be applied to the whole drift tube.