340ci vs 360ci engine
Unless you're going to run them in bone stock, factory configuration the 360 is probably a better bet. The 340 has a reputation for high performance because of the way it rolled out of the factory, with 10:1 compression, 4 barrel carb, forged crank etc. And the 360 never had that reputation because they never had better than 8.5:1 compression and many came with 2 barrels. But a 360 built to the same specifications will do just as well or better than the 340 (after all, 20 more cubes).
The rebuild on a 340 will cost more money. Not a ton, but more. There aren't as many piston options available as for a 360, and there are some quirks. Most of the pistons available for the 340's put the tops of the piston's above the deck .018", which makes it more difficult if you want to run an aftermarket closed chamber head.
As far as price, the 340's will cost more. I bought my 340 short block for just under $300 (a steal even though it was .030 over already), but I just bought a complete 360 that would probably run with a little work for $50. I've seen some crazy prices on 340's, but I've also seen quite a few in the $500 range for a short block.