Incorrect Tub Drain Installation?
I can't really tell from the picture, but the gasket looks kind of stretched out and I'd guess your leak is coming from it not sealing completely, not being located exactly right, or (perhaps because of the old putty?) the flange piece is not drawing down as tightly as it should to compress the gasket properly.
The gasket should be somewhat soft, elastic, and flexible in order to work right and ensure a good seal. Once you take the old flange piece off, if the gasket that has lost its flexibility (and they do over time) it just can't seal properly again.
You should remove all the old putty from the threads and fittings and clean it up as well as you possibly can. The old putty isn't really doing any good anymore and may be causing some kind of gaps or other interference that ends up leaking. A small wire brush might be useful to get all the debris out of the threads and so forth.
You don't want to really put the putty on the threads. The putty goes mostly under the lip of the flange and seals the gap between the flange and the tub. The gasket is what seals the drainpipe to the bottom of the tub and the flange piece needs to draw the two together to compress the gasket a bit and make it seal.