They want to access their apps on all their devices, but they don't care if all versions are built with the same technology as long as they can accomplish the task, at least i don't care as an app consumer.
Apps on each platform tend to look and behave in a similar fashion. E.g. on iPhone there's a tendency to use Tab bar buttons which are at the bottom. On Android you tend to have off-screen menu buttons and/or the Action Bar at the top.
Whether that's good or bad in terms of UX is a matter for debate but people get used to interacting with their platform in a certain way and it's disruptive to their user experience to suddenly find an app that doesn't follow these conventions.
For example, say you create an iPhone app where the menu buttons were at the top along the lines of an Action Bar you might find it creates a barrier to interacting with that app for a non-Android user.
So, this argument about UX really depends in which perspective of continuity you are considering, single platform oriented or cross platform oriented.
I have an iPhone and an Android phone.
When I pick up my iPhone and open an app I'm not surprised by an app using a Tab Bar interface.
When I use an app on my Android phone I'm used to options popping up when you press the menu button.
Yes, in a perfect world you'd have exactly the same interface on all devices. That's not today, not tomorrow, not next year and most probably never will be.
When you want to launch an app on your Mac you might go to the Dock.
On Windows, it might be in a Start menu.
On Linux it's somewhere else.
On a TV the menus are somewhere else.
Finally, there's native capabilities that HTML5 will always miss out on. It's why Java apps look and feel odd. Even though they try and mimic desktop look-and-feel (and often do an amazing job) it never quite works.