Why isn't the output of the LT1168 equal to its input in this very basic circuit?
You forgot to connect the reference pin of the in amp. This is usually connected to ground. Here's the block diagram of the in amp, taken from its datasheet:



With the reference pin open the difference amplifier (composed of \$R_5\$ through \$R_8\$ and the op amp \$A_3\$) does not have the normal gain of 1.

Add a ground connection to the reference pin (the one near the "LT1168" in your image) and you'll get a gain of 1.