I own a 156 1.8 Veloce, and have a friend with an IS200.
I haven't driven the IS myself, but have been in it enough times to say it is better than the 156 in most areas. However, it cost nearly 10k more than my 156, and it is only a couple of years newer (2003 instead of 2000).
Handling & Ride: For a FWD car the 156 handles well, but the RWD IS200 is in another league. It also manages to have a incredibly smooth ride as opposed to the harsh ride in my 156 Veloce.
Engine: Both have similar power, both have similar 0-60 times (just about 9s). The 6-cylinder in the IS sounds very nice. The IS does need to be constantly floored to get anywhere as it's so heavy, but so does the 156 really. The IS feels like it pulls much better at higher speeds than my 156 (possibly as mine is only the 1.8)
Equipment: Pretty much the same but the IS has some nice little touches like electronically adjustable seats, folding wing mirrors, 6-CD changer in the dash etc.
Interior: The IS has a nice interior, but I much prefer the 156. There seems to be more room in the 156 for passengers as the electronically adjustable front seats in the IS take up lots of room, and the transmission tunnel in the IS makes it difficult to fit someone in the middle seat in the back. Also the boot on the IS has huge hinges which take up lots of boot space, so the 156 boot is much better.
Running costs: Probably similar on insurance, similar on repairs, book figures for MPG are the same, perhaps IS is a couple of MPG less.
Overall, I'd say the IS is a better car in terms of more refined, better performance, but it's just not as fun as the 156. And you certainly pay a lot more to get one.