The LS is an S-Type down .1L on displacement (yeah, marketing gag), different suspension settings/tuning, and with a different interior. Ford claimed that almost no parts were common between the two... mostly because the ones they supplied had different serial numbers stamped on the parts. In reality, something like 70% of the parts interchange.
It was also probably the best Lincoln they ever made. Lincoln didn't know what to do with it, so when they did the facelift/update (think it was in 01) they softened up the suspension and tried to make it emulate the Town Car more. Idiots.
Per most of the motoring press, the DEW98 chassis that both cars were based on was too down market for the S-Type... and too upmarket for the LS. Having driven both, I'd have to agree - it was astonishing how good it was for an "American car."
I'd recommend the first-gen LS over the S-Type any day.
Bottom line, the LS was better at what it was supposed to be than the S-Type was at its niche. And, sad to say, the LS was put together better in addition to being a better value.
The Thunderbird was also on DEW98, but that was a Michelle Cervantes disaster - she dropped the plot and made it a total retro car, right down to the 50's handling.