The adapter sleeve is typically used when the bearing is on a shaft without a shoulder and needs to be placed properly in a pillow block housing for example. This sleeve uses a hydraulic or mechanical device or nut to tighten the bearing on the sleeve and hydraulic oil injected into the end of the sleeve on larger sizes to release bearing when removing.
The withdrawal sleeve is used when the bearing is installed against a shoulder on the shaft and pushes the bearing against the shoulder while tightening. The advantage of this is the bearing can be installed with the shaft out of the machine since it is pressing against a shoulder, with the adapter sleeve, it would be more difficult to position the bearing with the shaft out of the machine since it is free to move on the shaft until it is tightened.
The choice of which to use will depend on the design of your shaft and how it will be installed. The attached document shows the mounting and dismounting methods of both sleeves and will probably answer many of your questions.
For the withdraw sleeve you have to worry about the stress concentrations put into the shaft. Threads and a keyway have to be machined into the shaft for the sleeve.
