Hmm that's an interesting question. My entirely subjective impression is that "planet" commonly refers to the actual physical planet Earth, while "world" can be used more broadly to refer to the peoples of the world or to the broader idea of all the things that make up the concept of the known world. Hence "ON the planet," (physically on the surface of the earth,) but "IN the world," (part of the totality of things that make up the "world.") Presumably when people say "on the planet" they mean in a more physical sense, but when they say "in the world," they probably mean something more conceptual, since it would be decidedly odd to imply that something is IN the world, in the sense of being physically inside the planet, unless, I suppose, you're talking about the earth's core or something like that.
That said, we don't have to be subjective here. Take a look at this NGram, (linked below) and you'll see that, happily, the phrase "in the world" appears to have enjoyed greatly superior popularity since at least the early 1500s, and continues to do so today. Also, you can see that around the turn of the 18th century, it became super popular to capitalize the word "world." Of course capitalization rules were quite different then too. The word "planet" was also capitalized a lot more around the same time, though you can't see it on this NGram because the usage is so far below "in the world." Down at the very bottom, the phrase "on the planet," both with and without a capital P, was and remains far less common. You can see a clear uptick in use, however, at the beginning of the 2000s, and that's probably what you are perceiving. This, of course, deals only with written data, and not spoken data.
Here's another one showing just "planet" versus "world."
The Corpus of Contemporary American English, by contrast, contains a hugely varied range of sources for English utterances beyond literature and purely written sources. It includes both written and spoken English, and contains hundreds of millions of words. It documents the phrase "in the world," (capitalization is irrelevant this time,) a total of 44,787 times, compared to a meager 3,049 instances of "on the planet."
Searching just the word "planet" returns a more optimistic 24,683 instances, but searching just the word "world" still yields a much higher return with 382,531 instances.
I couldn't say why you may be seeing it around more in recent years--that NGram only documents up to 2008, and I don't know of any linguistic studies showing the relative popularity recently. However, I think it's safe to say that "in the world" is a long way from being usurped by "on the planet."
I hope that helps, and thanks for such an interesting question!