First of all, it's definitely So
chi (but sounds more like
TCH), this is official transliteration of the
Secondary, as others mentioned, you must do a stop during your flight at Moscow - unfortunately, in Russia many flights are made with such stop. I suggest you to stop for a day in Moscow, just for another sightseeing there.
Flight duration from Moscow to Sochi - 2 hours
Flight duration from Moscow to Khanty-Mansiysk - 3 hours, and flight by is made from other airport than you'll land from Frankfurt!, if you choose the UTAir.
You can choose the TransAero, but it's not as cheap as Sochi flight.
And most important part: sightseeing and weather.
Khanty-Mansiysk's average temperature is -9,8 °С, and sometimes there is cold weather,
down to -40.1 °С (March, 1st, 1966).
Sochi's average temperature is +15 °С, with sea temperature up to +9 °С. You can try it out - extreme, but quite fun :) There will be a snow machines for Biathlon in Sochi, but natural snow is in Khanty-Mansiysk.
As for the sightseeing, in Khanty-Mansiysk is all about the World Cup - people there love skiing, and you will be bored there if you are not a fan of it. There are quite small quantity of the sights in Khanty-Mansiysk (in Russian), and even less of them you'll be interested in during cold weather.
As for Sochi, you'll find much more fun there - because it is a very popular touristic center during long time. And because of preparing for the Olympic games, there are much more easier for foreign tourist to get information about city - people started to study languages, and whole touristic situation changed in a good way.
So, make your choice, but if I were you, I'd start with Sochi (plus 1-2 days stop in Moscow) - for start it will be enough. But if you want some extreme - choose the Khanty-Mansiysk, it will be a journey of the year for you.