They are not the same. A vinegar is basically a wine gone bad. The name comes from the French for "sour wine," and refers to the fact that any alcoholic beverage turns sour when it is exposed to air. This natural process turns the alcohol to acetic acid, and a vinegar is born. Before the advent of the wine cork, in fact, keeping wine from turning to vinegar was a constant problem.
Nowadays, making vinegar is a controlled process, in which certain bacteria are introduced to the alcohol (red or white wine, rice wine, must (for Balsamic vinegar), malt, fermented cider, or any liquid containing sugar or starch that is induced to ferment) and allowed to ferment further. In more traditional vinegar making, certain parasites were also added to work with the bacteria to promote superior flavor (but we won‘t get into that).
Getting back to your recipe, rice wine is a sweet wine made from fermenting steamed glutinous rice, which has a quite low alcohol content. The better known rice wines - sake and mirin - come from Japan, but there are many Chinese varieties, too.