Knead 'Tang Yuan' 搓湯圓 and Shake 'Yuan Xiao' 摇元宵.

A brief note on what *I think* differences between Tang Yuan 湯圓 and Yuan Xiao 元宵 - just for the record.

Historical facts: 
Yuan Xiao - it's easier to explain Yuan Xiao by its Korean name 正月大滿月 which means Big Full Moon. Yuanxiao therefore means the Evening/Eve of Big Full Moon (to Koreans). 
Yuanxiao  - the yuan 元 therefore is about the Moon (Lunar year, welcoming Spring). It's about a religion. It's a public event.
Tangyuan - the yuan 圓 is about family reunion. It's a more culturally homey, private event. 
Therefore - Yuan in Yuanxiao and Yuan in Tangyuan is NOT to be confused. 

Yuanxiao 元宵 is in a Ming dynasty literature as a Northern China term equivalent to Tangyuan South of The Yangtze River.
The process of Yuanxiao is detailed in this Ming literature - that is to shake (which means roll not by hands, but by in a basket). Hence the term 摇元宵. The reason I call it a 'public event' is because it is for mass production, and the basket is conveniently replaced with machines when they became available). There is no family reunion significance in 'shaking' yuanxiao. 
Tangyuan 湯圓  sounds similar to  Tuanyuan 團圓 meaning family reunion. So there you see what I mean in the 9th sentences above. It is documented the name of this sticky rice ball is simply called 湯子 (the food in the soup) in Song Dynasty. So who started calling it Tangyuan to connect to Tuanyuan (family reunion) - don't know. But that's how it is today. Either in China or Taiwan. It is traditionally made by hands, called 搓湯圓 Knead soup ball. 
One is roll by shake. one is roll by knead.

BOTH traditionally is made a SWEET dish - as it is served as a dessert (or better SWEET TREATS, since China don't do dessert course - an after-meal culture) and symbolise a sweet, smooth forthcoming year, future. Savoury filling therefore is modern variety - none traditional. 

So that's historical facts out of the way. 

Now, for modern days Taiwan Yuanxiao and Tangyuan
(and from my personal experience and observation, that is)

THE SOUP: The process of making the two naturally makes soups of different thickness - Yuanxiao rolled and shook in rice flour resulted in much more evenly rolled moon shape ball. Coated with rice flour naturally makes the soup thicker. 

Tangyuan, spare of the flour coating, naturally is served in a clearer syrup. 

My experience tells me, in Taiwan - where in modern days, family don't sit around tables  much preparing food including rice balls. And there is never shortage of range of flavours and fillings to cater preferences and occasions (such as whether eat it savoury as a 'meal' or sweet as a snack or dessert - like served at the end of a banquet with good wishes of sweet, smooth, rounded future. or like Hakka savoury filling tangyuan I would order one for a meal)

In Taiwan - tangyuan is traditionally none-filled. whereas yuanxiao is traditionally sweet-filled and eaten around this time of year. 

For culinary aspects - therefore, yuanxiao in Taiwan is decisively a sweet soup - due to its sweet filling.
Tangyuan, with no flavoured filling, is a neutral bowl of soup. One can eat it sweet served with sweet soup, or eat it savoury served with meat, vegetarian fillings and thrown in greens and eggs to make it a savoury meal. 

The video is in Taipei Street - 'Shaking Yuanxiao' and sold in boxes.