I'm researching this now for clients and here is where I'm at as of now.
QB Desktop: Intuit used to have a mobile app for QB Enterprise (all the desktop products) in 2018, but they pulled that in favor of the App store app for QB Online only. I'm looking at the QB Desktop Apps section another poster pointed to and there is an app called Invoice Tracker that syncs with desktop, not by Intuit, but by a 3rd party. I have not tried it myself.
QB Desktop Hosted: This is when you pay someone to host your QuickBooks Desktop software so as many users as you pay for can get anywhere, anytime access, the way one would with online, only with the advantage of the better quality QB Desktop software (including Enterprise). One client uses a host that we access using RDC (remote data connection). I can get access on my PC and my Mac, but does that lend itself to access by an iPhone? Another client is using a host that uses Citrix. I can access that one directly through Chrome, but does Citrix work well with the iPhone and Safari? When researching a host if iPhone (Android) invoicing is important, I would make that a primary question, and test it before committing everyone in accounting to that host?
QB Online. This is the product Intuit wants to sell because of the profit margins. When one goes to the App store and see QB Accountant, this connects with QB Online. I was at a client's last week for whom it was a big deal and he was super happy with how it worked. But then in my opinion QB Online is overpriced, slow, and buggy software so it's a trade off. For example, that same client has been overpaying salestax because the salestax feature & reports don't work for love or money.
I'm still looking for better information? Once I get it, I will update this blog article with better information and add it to my white paper on "Remote Access Options for QuickBooks". If you have anything to add or recommend, please do. This is an important topic.