This must have been discussed bazzillion times but as owner of both I'm glad to chime in.
Generally TOUGFC said it all.
Couple of things from my experiences:
DA40 focuses faster (almost much faster)
DA40 has very typical DA rendering, warm but neutral, great contrast, kind of brutally honest
DA40 shows less CAs in contrasty situations (visibly so in certain specific conditions)
DA40 is smaller/lighter (almost half the size)
DA40 is sharp edge to edge
I've never seen jagged circular OOF from DA40 my FA43 does do it sometimes
DA40 is generally not as picky with background for delivering smooth OOF
FA43 has much warmer (yellowish) cast (I call it glow and so far I found out it's typical for all FA line lenses I've owned - *24, 31, 43, 77, 100/2.8 Macro, 135)
FA43 is visibly sharper in the center of the frame
FA43 for some reason gives more 3D look (or more pop to images as some say)
FA43 has weaker corners but by f4 it's as sharp in the corners as DA40 IMHO
FA43 is relatively slow to focus but has much better MF feel
I much prefer cap of FA43
FA43 has bigger resale value (especially if it's MIJ copy)
FA43 (at least my copy) seems to be less prone to PF (this varies wildly across the copies as I've seen images with quite some PF but even the worst cases were at least comparable with DA40)
FA43 feels bit better build (somehow) than DA40 (my copy of DA developed slight barrel wobble after 6 months or so...)
FA43 can have better OOF but user needs to keep an eye on the background otherwise the OOF can look quite nervous....
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Bottom line, they are both great lenses. I used to love DA40 but the f2.8 was bit slow for my liking, 43 is pretty much all I called for but doesn't get used too much since I got K50/1.2....
good luck choosing