Do the levels depicted on the Jungle Book box art exist in the game?
Yes, they exist in the game. But not in the version of the game you own.
The Jungle Book game is an interesting case of some weird things going on. The game was released for NES, SNES, Genesis/Mega Drive, DOS, Windows, Game Gear and Gameboy and while all versions follow the same story, not all versions are actually the same. In fact, the NES and "mobile" ones are limited due to the nature of those consoles, and the SNES one wasn't even created by the same company.
To see some of the major differences, here's a playthrough for each of the versions of the game.
- NES
- SNES
- Genesis/Mega Drive
- DOS
- Game Gear - Important Part of Video
- Gameboy
You can see all of the game versions vary and sometimes to an extreme degree. But notice the call out on the Game Gear one. The time in the video I linked to in that playthrough is the same screenshot as seen here.
You'll also notice that between some of the screenshots on the back of the box, the UI is different. This is because they are coming from different versions of the game.
So, in short, yes, the screenshots exist in some form in one of the versions of the game you own. But it is not the version in the box. It looks like the box art team did a bit of mix and match for the source of the screenshots which happens from time to time. But usually the versions of the game aren't as drastically different as they are in this case. This just happens to be a special game.