I've owned multiples of both, though all my JBs have been ruggeds and not custom leather cases.
Whitten makes a very nicely built case that looks incredible when new. I think they're a great storage case at home but I really don't love playing out of them. First, the leather covering is pretty thin and tens to wear poorly in my experience. You can find pictures of scuffed up Whitten cases around and they don't look nice because they use thin heavily finished leathers. The tubes also allow the cues to move around more than I like the the pockets just aren't great to play out of in my experience. Finally, they have metal feet and are very slippery on non-carpeted floors so I find them less stable when leaning them in the corner.
JB cases are also great quality. They provide the ultimate in protection, though personally I wish the fit was a little less tight on a fully loaded case. But, I just always buy them slightly larger than I need and they work great for me. I've run the 3x4 as a 2x4 and that is perfect and I'm presently running a 4x8 as a 3x6 and it's also great and I can squeeze in another butt if I really want to. Honestly, they'll hold the full capacity, but they do get pretty snug. JB cases have a great pocket design that is really functional and a full rubber bottom so it never slips.
For me, JB wins all day long as I just find them so much better to actually carry and use on a regular basis. They're really well thought out designs for a player and not just made to look pretty.