I have over a thousand hours in PA-28s of nearly every version, including Warriors, and about 250 hours in Mooneys, including nearly 20 hours in the past month flying a 1967 M20F from Seattle to SoCal to Phoenix to Dallas. So I think I'm in a position to address this question.
Everybody talks about "cabin width". Mooney compares well in that category, as shown in this Mooney ad from the 1970s (interior dimensions in an Arrow are the same as in a Warrior). Note that the Mooney's width is measured at elbow level at a small cutout in the sidewall; overall the difference in width is negligible.
But cabin width at the elbow is only part of the story. The upper half of the Mooney's cabin is more rounded than the Piper's, and it seems a little tighter at shoulder and eye level than the Piper. And if your older Mooney has the original windshield, you'll certainly notice how close it is to your face. The '67 I flew had been modified with the 201-style sloped windshield, so it was fine.
The Mooney's cabin door is smaller than the one on the PA-28 (especially the post-1972 PA-28s). I'm not as limber as I used to be, and that door makes a difference. Once in the right seat, you now have to slide over to the left. The Mooney has a huge nose wheel well housing between the footwells, and getting around it with long legs is a challenge for the pilot and fun to watch for everybody else. Once in the left seat the seating posture is comfortable enough, though lower to the floor than a PA-28. The seat cushion of the '67 Mooney I was flying was uncomfortable, but I think that was just an issue with the reupholstery job on that particular airplane.
I like flying Mooneys, this '67 model included. But at my age the interior of a PA-28 suits me better.
Just my $.02.