I have read his Civil War books about Franklin, Shiloh and Vicksburg and that is the order I would rank them. I read Shrouds Of Glory twice. Only Civil War book I have read twice.
I have read the same three and would rank them the same. I've also read his book about New Orleans (1815 not 1862), which I would definitely rank below the other three and would hesitate to recommend. Groom seems to generally understand the Civil War; the War of 1812 not so much.
Shrouds of Glory (Franklin & Nashville) was one of the first adult Civil War history books I read after Foote - mostly because my local library had a copy. (Why I didn't read Catton I don't know.) It is also one of the few Civil War books I've read twice, but mostly because I'd originally read it in 2004 then revisited it over a decade later once I had started reading about the war much more extensively. I haven't read any other books about Franklin or Nashville, so unlike Shiloh or Vicksburg or New Orleans I can't spot the issues as well. (There's a CWT member who despises that book with a passion.)
However,
Shrouds of Glory came out in 1995, when Groom was new to nonfiction and it was the only history book he wrote that decade. He could take his time on it. In contrast, from 2002 to 2020 Groom was cranking out history books (many not about the Civil War) every 1-2 years. Few can produce high-quality books at that rate, especially at the 300-400 page length.
Groom is a good, enjoyable writer but a weak historian. He accepts sources uncritically (in the Vicksburg book especially) and makes some careless errors, like conflating Thomas ap Catesby Jones from the War of 1812 with Catesby ap Roger Jones, commander of the CSS Virginia in 1862.
If someone doesn't normally read books about the Civil War, I'd recommend Groom's book about Shiloh over Smith, Cunningham, Daniel, Sword, or McDonough. But anyone interested enough to be on CWT should read Timothy Smith instead of Groom.
Comparisons to Shelby Foote in terms of style are apt, but I think Foote was more diligent in his work.