A1: I am confused by the usage of Analysis Process Designer (APD). What is it used for?
A2: It seems that the Routine in Transformation is an alternative to APD, is it true? What is the difference between the APD and the Routine?
It's late here and I don't feel like going into too much (obvious) detail, but I stumbled across your question and want to provide at least a partial answer, though I doubt it comes in time seeing that your question is already a few months old.
One neat feature of APDs is that you can specify queries as source. This is not possible with transformations. Using queries as source is particular useful if you're facing an Inventory scenario per SAP recommendation, i.e. a scenario with non-cumulatives as opposed to a snapshot scenario, as the query calculates the stock balances automatically so you don't have to.
Other than that, APDs can be used for data mining, which I'm not familiar with under SAP BW, as well as a few other things that I also haven't had any experience with. Unless I'm much mistaken, SAP no longer develops the APD and there are quite a few bugs that I'm aware of, even in the current versions of BW.
At any rate, depending on your use-case, you'll most likely be perfectly happy with sticking to transformations and their start, end, and expert routines unless you have a very specific scenario or requirement.