My library system, Hennepin County, has a mobile app that seems to be both popular and a source of complaint from our users. I have it installed on my mobile device, but I have to admit I rarely use it. Until today I was never fully conscious of the reason, but it just came to the forefront of my brain: it's because on the full website I can store all three of my family members' cards and easily toggle between them. On those mornings when I'm running around the house scrambling to find all of the books that are due that day, the full website -- even on a mobile phone-sized screen -- is vastly more helpful. Since that is the only situation in which I access library accounts on a mobile device (I do everything else from a desktop computer), HCL's mobile app holds little appeal. Having said that, I very much appreciate that the full website is easily navigable via my iPhone or iPad.
I haven't used any of the vendor apps. As above, I take care of nearly all of my library business in a non-mobile setting.