Is this better because your not running the high the current through the transistor, and therefore can use a cheaper cooler running component?
Yes, that is the main reason from an endpoint perspective. In fact, a 2N3904 is even overkill with respect to specifications, but it is probably the cheapest transistor in the world.
I am interested in knowing the reasons for the design.
Using the adjust pin is how the design engineers intended the device to be controlled. The device already has the ability to do the 'heavy lifting' by design, so why add a big, hot transistor to it to get the job done.
Unfortunately I am unable to test until Monday, when I can get hold of the additional parts.
From a total parts perspective, it really is only one additional resistor (the 1K). The caps should be there in your circuit as well. (well, in the corrected one suggested by Erni).
Also I am confused about which part sets the the LM317 - I thought it was a resistor 1.8, between ADJ and OUT, but here its quite different!
Not really. It is still the 1.8 ohm resistor that sets 96% of the 1.25V control voltage. Because the adjust input has a very high impedance, the 1K resistor (which is there to limit current through the transistor) has negligible effect. The current through the adjust pin is only about 50uA. So, doing the math, it introduces a -4% error in output current which is well below the tolerance of the rest of the circuit.