The difference between balm vs. liniment is that most liniments are alcohol based, whereas most balms are petroleum or natural oil-based. This is an important difference in whether you want something that is liquid or solid (for transportation, application, etc.) So, you should first figure out whether you want something that is the consistency of Tiger Balm or if you are open to other forms, such as liquid, gel, cream, etc.
If you are looking for a balm similar in texture to Tiger Balm, here is one really great alternative. One of the best alternatives to Tiger Balm is a new U.S. manufacturer of a petroleum-free, all natural balm that contains quite a few Chinese herbal ingredients. It’s called Battle Balm and its uses are more than just the typical muscle rub and topical analgesic. It is specifically formulated for contact sports and trauma associated with it. The Battle Balm website says that it can be used for sprains, strains, muscle pulls, tears, tendonitis, arthritis, and small fractures. There are also 17 other uses for it. Even if it is not exactly what you want, at least the Battle Balm website may help you figure out what kind of product is right for you.
I hope that helps.