First, it's a *lot* slower than driving.
On the NEC, of course, the Crescent is faster than driving; and from Washington DC to Greensboro NC the Crescent is only a little slower than driving (5:45 vs. 4:45). From Greensboro to Atlanta it's substantially worse (7:34 vs. 4:55) but it has the benefit of overnight running, and of course it's mostly through trips headed north of Greensboro.
But south of Atlanta, it's *much* worse. Atlanta to New Orleans is 11:30 vs. 6:38 driving.
The NS tracks from Atlanta to New Orleans are part of their Crescent Corridor route. The Crescent Corridor is the subject of a $2.5 billion improvement program with some of that money coming from the feds and states. The question is whether the Atlanta to New Orleans segment is slated to get upgrades that will allow the Crescent train trip times to improve south of Atlanta. Not by a lot I would expect, however.
The NS tracks north of Atlanta will see trip time and reliability improvements in the next 3-4 years with the Piedmont corridor Greensboro to Charlotte upgrades and VA paying for a double tracking project and likely other upgrades between Manassas and Lynchburg for the Lynchhburg-Roanoke Regionals.
The NS Crescent Corridor map is below for those not familiar with it. If Virginia pays for passenger service upgrades to Roanoke and then Bristol, VA, the map shows the possibilities for a NYP to Memphis train if TN was someday interested in helping to fund it.