In this specific case, comparing the LTD to the Edwards...
1. The Edwards has the true RR (Japanese) body shape compared to the US version with the larger lower wing.
2. The Edwards has the ESP paintjob (sexy).
3. The Edwards doesn't have neckdiving problems (not a huge problem on the LTD as you can have the strap button re-positioned or use a leather strap, but it's still a nuisance nonetheless).
4. The Edwards has scalloped frets from the 20th to the 24th fret. I've personally never experienced scalloped frets, but they sound nice.
5. The Edwards has the MM-04 pre-amp booster built in. Not sure how good it sounds, but it is a nice feature.
Essentially, the Edwards has the exact same "specs" as the ESP Custom Shop version, minus the build quality. It is a bit expensive at the moment, but if the price drops I'm seriously considering replacing my LTD with it.