What difference between two discharge resistors in two circuits?
Circuit 1 will light the LED on one half cycle only. This may result in some visible flicker. Circuit 2 improves on this by using a full-wave bridge recifier to light the LED on both half-cycles.

There's no real effective difference for the R1 connection. In Circuit 1, C1 is discharging through the load, and in Circuit 2, it's discharging across the cap directly. That said I'd prefer the way it's done in Circuit 2, as it localizes the discharge loop.