340 or 360?

C'mon, people.
If you're doing a build, it basically don't amount to a hill of beans if you start with a 340 or 360, as long as the basis is sound. What counts is what parts and how much thought you put into it, rendering any comparison of the two pointless.
For factory engines, the 340 was never anything but a performance engine, with a relatively short production life; '68- '73. Similar to the street hemi, limited production life, one dedicated purpose- performance. There was no grocery getter version of the 340. In all it's years of production, I can only think of 2 deviations from the yearly long block production norm: the 340 T/A and AAR E bodies, and a reverse-rotation 340 for Chrysler Marine for twin-engine installations.
The 360 was in production from '72-'92 (2001 if you want to include the 5.9 Magnum) and powered everything from Dusters to Winnebagos, generators to irrigation pumps; in a thousand versions including 2bbl., 4bbl., TBI, MPI, natural gas, flat tappet, roller tappet, industrial spec, Mil. spec, police spec, marine, and on and on... Lets face it, these things were EVERYWHERE.
So which do you think is going to be more popular? The 340 is the performance legend that you have in mind as you build your 360. Sometime the legends get embellished over time, but that's the nature of things. The 340 was intended solely as a performance "image" engine, and it's successor, the 360 had much broader intentions for it's use. Two engines born in 2 different eras to suit different intentions, but either one can be a viable performance engine for a multitude of applications; but as I said before, that's all in the build.