Constant current circuit with transistors/MOSFETs design comparison
The two are really the same, functionally. Both work by regulating the voltage over R2 to about 0.6V, what it takes to forward-bias the base-emitter junction of Q1. If the voltage over R2 increases beyond this, Q1 begins to pull down the gate/base of the other transistor. But it can't do this too much, else there is no current in R2, and nothing to forward-bias the base-emitter of Q1. Thus, the circuit achieves equilibrium.

The idea is then that since the LED and R2 are in series, their current is equal. If you can make 60mA in R2.

This is only approximately true, of course, because R2 and the LED aren't exactly in series. In both cases, errors are introduced by the base currents of either transistor. Fortunately, the current gains are very high, so these errors are negligible. I doubt there is any practical difference between the circuits, so selecting based on what you have on hand sounds good to me.

However, if your goal is 350 mA in the LED, then R2 needs to be \$ 0.6V / 350mA = 1.71\Omega \$. You might want to use a 1/2W resistor too, since you are pushing your luck with a 1/4W: \$ 0.6V \cdot 350mA = 0.21W \$. Be sure the transistor you select for Q2/Q3 or M1 can handle the power it must dissipate as well, which will be 12V, minus the 0.6V in R2, minus the forward voltage of your LED, multiplied by 350mA.