Much like the term otaku, yuri (although originally a Japanese loanword) has undergone significant semantic drift. In Japanese, the term is typically used to mean any lesbian content, whether sexual or romantic, explicit or implied; the term shoujo ai is not found in this context. American usage has historically differed: 'yuri' has typically been used to denote only the most explicit end of the spectrum, being effectively a variety of hentai, while 'shoujo ai' (independently coined, following the most common use of shounen-ai) described anything without explicit sex. However, American use of 'yuri' has broadened in recent years, picking up connotations from the Japanese use: the precise difference between 'yuri' and 'shoujo ai' -- or even whether or not there is a precise difference -- will vary depending on the speaker, and is rarely objective in any case. Common definitions include:
Yuri contains explicit sex; shoujo ai does not.
In yuri, the lesbian relationship (whatever its nature) is explicitly shown; in shoujo ai it is only implied.
Yuri is passionate; shoujo ai is romantic.
Yuri is ecchi; shoujo ai is innocent.
Yuri is a general term; shoujo ai is "fluffy" yuri.
Yuri and shoujo ai are equivalent.