Why is it important to keep the stem while processing the tea?
There are two main kinds of Catechins present in the stem of the oolong plant. The first being epicatechin and epigallocatechin. The second being , epicatechin-3-gallate and epigallocatechin-3-gallate. The former type is "free form" or "simple catechin", while the latter type is "ester form" or "complex catechin".
These two different kinds of catechins each create a different feeling on your palate. The simple catechin will be less astringent and is present in higher quantities in the the mature leaves and "stem". The complex catechin is more astringent and is present mostly in the bud or immature tea leaves. The crucial reason to keep the stems while picking the tea leaves is that during the process of indoor withering the simple catechin in the stems can be distributed to the leaves not just to make the tea less astringent, but also assist the "aroma molecule" inside the leaves to be released more easily, which helps (later) during the oxidation process to create a better more complex tasting tea.
Stem is an important part of tea processing, do not miss it!
So with that concept in mind, let's talk about the Anxi no-stem and Taiwan with-stem phenomena. How do they differ and why?
It's really a matter of style. The modern low-oxidation "green" Anxi Tie Guan Yin is picked with the stem and processed with the stem so that the afore-mentioned catechins will be imparted by the stem. Then after processing is completed, sometimes days or weeks later the stems are hand-separated (挑梗). Since modern-style "green" Anxi Tie Guan Yin is appreciated for it's strong aroma and green taste the stem counters that with a mellow sweetness which is aesthetically undesirable.
However, the traditional Anxi Tie Guan Yin is processed with 50% oxidation and heavy roasting techniques requires the stem be retained even after processing is completed. There is a saying in Chinese that reminds us the traditional form oolong tea should take; "toad skin; froggy leg; dragonfly head, Rice Dumpling leaves shape" (蟾蜍皮，青蛙腿，蜻蜓頭，粽葉蒂). The froggy leg part indicating its form of having "stem".
The modern very green Anxi Tie Guan Yin is processed using an "air-condition" withering instead of the traditional "sunshine withering". Additionally modern Anxi processing techniques don't use the traditional "tea-shaking" (浪菁) step anymore, so while the tea has a very high level of aroma it doesn't have the body that is required for the traditional Tie Guan Yin (which includes the stem).
In conclusion, the tradition of keeping the stem in Taiwan has been retained. The roundness and sweetness imparted by the stem is enjoyed from start to finish. Modern Anxi Tie Guan Yin is processed to the finished product with the stem, but then the stem is removed before drinking so that it's mellow character doesn't interfere with the pungent TGY aroma and vegetal green taste. In other words, it's a matter of style. Just as it's hard to imagine a high grade Li Shan oolong without the stem, it's also hard to imagine an Anxi Tie Guan Yin with stems!
We will still find stems from some traditionally processed Anxi Tie Guan Yin.
Taiwanese oolong still has stem picking process, but it is not as heavily picked as Anxi.