They use the same character, 元. That's it. It's a common character, meaning "round" or "round coin".

To elaborate, the Yuan dynasty was a period of Mongol rule over China, from c. 1260 (de facto, 1271 officially) to 1368 (1380 if you count the Northern Yuan dynasty when the Mongols retreated to Mongolia after Kublai Khan's death and subsequently being driven out of China.

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The Chinese Yuan, referring to the currency, 圆 formally but commonly the above character, is the basic unit of Chinese currency, much like the dollar is to many other currencies of the world. It was first minted during the Qing dynasty between 1888 and 1898 in various provinces.

While the modern rendering is officially rénmínbì (人民币) or People's Currency, in daily use 元 is far more common, or kuài (块) or "piece" in certain parts of China.

References:
http://afe.easia.columbia.edu/mongols/china/china.htm
http://www.allempires.com/article/index.php?q=The_Mongol_Empire

http://www.kenelks.co.uk/chinese/chinesedragon.htm
http://blogs.wsj.com/marketbeat/2010/06/21/yuan-or-renminbi-whats-the-right-word-for-chinas-currency/

Edited for formatting.