While listening to the Jim Lauderdale track, you should hopefully become aware of the feeling that the players are "holding back". It’s a hard thing to describe and quantify, but it’s what separates the men from the boys in the area of musicianship.
By playing just a little "late", a hair "behind" the middle of the "pocket", a great deal of musical tension is created, a feeling of anticipation. When that tension is finally released, it’s SO satisfying! That tension-and-release is very sexual, if I may be slightly vulgar. The tension is also maintained by the musicians refraining from playing any superfluous notes. As the old Jazz guys always said, the notes you don’t play are as important as those you do. For you Rockers, think of "I Can See For Miles" by The Who, and "Skakin’ All Over" by The Guess Who (The Who’s version on Live At Leeds is all about release, missing the tension created by The Guess Who. Compare the two versions!).
This style of playing is referred to by some as laid-back (often said somewhat pejoratively by those who don’t understand or appreciate it), and is for some reason a specialty of southern U.S.A. musicians. That is why Dylan started recording in Nashville in 1965, and why Jerry Wexler took Aretha, Dusty, Wilson Pickett, and other Atlantic Records singers down to Muscle Shoals in 1968-9. Wilson said he walked into the studio and saw these white crackers sitting around, and thought to himself, "Jerry, what have you got me into?" He says then the band (known as The Swampers) started playing, and he couldn’t believe what he was hearing---the funkiest band he had ever heard! The drummer, Roger Hawkins, is a favorite of mine (and of Jim Keltner, who says he wishes he played more like Roger) who was enticed out of the studio by Steve Winwood for a stint in Traffic. As good a musical drummer as I have ever heard.
Jim Lauderdale has that kind of taste in musicians.