Xbox 360 is supposed to continue alongside the Xbox One. It sold well in the US in the holiday season, according to NPD, and I'd assume it still appeals in many parts of the world where the Xbox One is just too expensive.

Lack of compatibility is an obvious drawback but consoles usually are incompatible.

I did just that. Although I bought a pre-owned console.

Given the huge (and growing) game library on the 360 that doesn't work on the Xbox One, it's hardly obsolete. I have two friends who just bought their first 360 in the last few months. They bought it over the One because (1) WAY more games (2) less expensive games, and (3) all our friends have one so lots of friends to play online with.

To echo this point, there are many people who take the PS2 approach of buying a console at the end of its lifecycle to take advantage of buying all used games or old games and rock bottom prices.