I have tried both of these products - they're both good but I will say that the BetterDry Day's are much better than the Tranquility ATNs.
The BetterDry's have much taller elastic leak guards to prevent leaks, and are way more absorbent than ATNs. I can easily wear and wet 3-4 times in a BetterDry without being worried about a leak; if I add in a booster and I can wear and wet comfortably all day and only change once around 4-5pm and then again before bed if I wanted to wear all day.
With a Tranquility ATN I feel like I need to change after two wettings, I wouldn't risk wetting a third time, I'm fairly sure they would leak. Even with a booster I wanted to change after 3 wettings and would have to change 3-4 times a day if I wished to wear all day. If you only like wearing for 2-3 hours at a time and just wetting once or twice? Maybe ATN is fine for you? Your call!
Some AB/DL folks like diapers that swell with a lot of SAP and the Tranquility ATNs definitely do this a lot! They start off fairly thin and discreet but after they're wet they get very swollen and 'squishy' from the SAP. Everyone is a little different on this, I'll take SAP/squishy over clumping but not if I feel wet which I often do wearing a Tranquility ATN so not my preference here, if I didn't feel the wetness that I do in an ATN I would prefer this over the BetterDry because the BetterDry Day's can clump sometimes. I find that if you 'activate' the BetterDry's by folding them in half and 'fluffing' them slightly this is less noticeable but might still be noticeable after the first wetting but goes away the longer you wear/more you wet.
Discreetness wise I prefer the BetterDry's. The plastic of the Tranquility ATNs is a little on the thinner side which results in them being a little more crinkly, too much so for my taste but some people may like this?
Tapes on the ATNs are pretty bad - they're thin but give you two tries. The BetterDry Day's are a little better, wider but not like huge, and again two tries.
So overall I think the BetterDry Day's are much better. Way more absorbent! But the Tranquility ATNs are ok for shorter periods of time but the absorbency and plastic backing is not really my favorite.
I'll toss in another diaper for you to consider picking up a package of in the future to try since they're kinda between the BetterDry Day and the Tranquility ATN - try the Molicare Slip Maxi's. Solidly between both products but I really like them!