I think both can be beneficial, I can really only speak from an Athletic Training standpoint as that was my major. The biggest pro for my major I believe is the clinical experience you gain as an ATC. As an undergrad, my program required 135 hours per semester from sophomore through senior year. That hands-on experience meant I graduated with at least 800 clinical hours(probably more because we all loved working our sports enough to go work when we're not scheduled), so I was ready to go out and be the health care provider assessing/rehabbing injuries on my own. The past 4 years between undergrad and now, I worked as the ATC at a local high school for a couple years, then was transitioned into the orthopedic clinic I work for where I still do outreach ATC stuff in addition to various roles around the clinic. I felt my athletic training background has really prepared me for the clinical side of PT. I still have a TON to learn in the classroom, but I feel confident in a lot of my hands-on stuff. I think highlighting these experiences are what helped get me into school as my grades weren't nearly as high as many of the people posting on here.
From what I know of my friends that majored in Exercise Science that were going the PT route, I think they went more in depth in the classroom than we did in areas like kinesiology and biomechanics. I think more research and projects as well from what I recall. So I'd imagine that in the academic sense Exercise Science may be a little more rigorous- from what I remember, courses like Bio/Physics/Chemistry were required for ES majors, while for me as an AT major I had to do anatomy/phys and then various other sciences, but not the aforementioned big three.
I also feel that Exercise Science, unless you go the PT/PA route, I feel it's more limiting than AT. At the very least, if I had never gotten into PT school, never gotten into grad school, Athletic Training is a career path I could've stayed on. I could've gotten my Masters in AT and worked at the college level, could've gotten into teaching AT, etc. I'm not as knowledgeable about Exercise Science so I don't want to step on any toes, but from what I can tell, if you don't get into grad school, it's a little more challenging finding a full career path with an Exercise Science degrees. I fully admit I've never done the research of what people with ES degrees do as most of the people I know in that field have gone on to PT/PA/some other graduate healthcare field so I'm sure I'm missing a lot of possibilities that an ES degree offers, again, don't want to offend anyone.
Ultimately, I think both are great preparation for school and definitely each have their own pros and cons. I think it boils down to what interests you more. At the end of the day, PT schools are going to teach everyone the same way, regardless of major. PT schools will teach everyone what they need to know, so I don't think picking one over the other as an undergrad will turn into a "Man, I should've chose the other one." situation later.