Perilla Oil is the term used to refer to any oil derived from the plant perilla frutescens, and it almost always refers to an oil that is pressed from roasted seeds which is used in the diet as a condiment to confer a nutty taste. It is sold as a dietary supplement or 'health oil' due to having the standard phenolic and Vitamin E content that many 'healthy' oils do, but is also quite a large source of omega-3 fatty acids (in the form of alpha-linolenic acid; the plant based form not to be confused with fish oil) comparable to flaxseed oil.
The leaves of perilla frutescens are less frequently used, by are sometimes used medicinally for the purpose of easing the stomach (served alongside some seafood); the oil pressed from the leaves, which is much less common, may have different properties than the seed oil and the two should be seen as somewhat different supplements.
Perilla Oil appears to have some promise associated with its rosmarinic acid content and the omega-3 fatty acid content, but the unique properties of perilla oil are not well researched nor is there much evidence on the benefits these isolated molecules confer. The benefits of perilla oil can possible be mimicked by flaxseed (if the omega-3 fatty acids are what does the benefit) or either supplementation of pure rosmarinic acid or plants that have a higher rosmarinic acid content (such as lemon balm).