So... judging by the comparison that you made of American support for the Mujahideen in Afghanistan, I suspect that you aren't necessarily aware of the differences between those conflicts. Afghanistan was an independent nation, which was 'invaded' by the USSR (or invited by the government, depending on your perspective, but being its own country is the important part here). The United States, along with other countries, was providing backing to a resistance movement attempting to expel the Soviet troops.

Chechnya, however, was a constituent part of Russia (whether or not it should be depends on your perspective, but the fact Russia considered it to be is the important part here), and the conflict was over separatism as they sought independence. The Chechens saw themselves as expelling foreign invaders, perhaps, but that was not the situation from the perspective of international relations.

So, the core difference is that while the US could get away with supplying aid to the Afghans who were fighting in Afghanistan, if the US started backing Chechen separatists, that would be basically an act of war, as Russia would consider it meddling in internal affairs, basically the same as if Russia started backing an armed separatist movement in, say, North Dakota.