When I'm cooking, I actually notice the difference the most. At home, I use plain yogurt for marinades and sauces because it's just more stable when things heat up. Regular yogurt with starch and thickeners in it - that keeps breaking on me whenever I heat it. It's really annoying when you're making a sauce and suddenly it looks like flake soup.
Taste-wise, there's definitely a difference too: plain yogurt just tastes like... well, like yogurt basically - sour, intense. Regular yogurt is often sweeter and creamier because of the sugar and starch they add. If you want to eat it completely plain, plain yogurt is probably the better stuff because you know what's in it. But I'd also recommend plain yogurt for refining dishes, just because the consistency stays more reliable when you heat it.
Legally speaking, the previous answers are right - there's no strict definition, it's more marketing nonsense. But practically in terms of processing and taste, you definitely notice the difference.