Is it more important to
- keep an application layout faithful to the native design/visual of a platform
- customize it to look like your own app?
Example: You could change an Android native dialog layout/colors, to be more like the app design.
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I'd say "both" because "look" and "layout" aren't quite the same concept.
Customizing colors, typography & interface elements is common & pretty much expected to help make apps stand out and not look so "standard." Change UI colors & typefaces to match your branding, add a logo or custom controls, etc. If every app looked the same it'd be pretty dull.
Layout, however, has more to do with where you decide to put things, what elements you choose to emphasize (text vs images, etc?), and the expected navigation/interaction design patterns. These decisions should be based on what's best for displaying/interacting with your app's content. If standard patterns don't work for you of course you could come up with something entirely unique to your app, but try not to add too much to the learning curve for the user.
Where problems might arise is if you customize controls, gestures, or standard patterns in a way that overrides expected behavior, or if you create a unique substitute for common functionality.
For example, app users have become accustomed to "swipe to delete" or "pull to refresh." If you decide that your app will use "pull to delete" instead, that will probably confuse & anger people when all their data disappears.
People also generally expect certain functionality or system features to appear in certain places and to look/act a certain way, based on what OS you're using (ex: the iOS Navigation Bar typically puts the "Back" button at the top-left). If you decide to move things to unexpected places, or use non-standard icons to represent them, you also risk confusing & frustrating people.
In that sense, "layout" customization is very different from "look & feel" customization..
I would say the most important thing is consistency. If you customize the appearance, you need to customize it everywhere to provide a consistent experience to the user. If you decide to go native, you have to go native everywhere to provide a consistent experience to the user.
I would vote native, so that the user has as consistent of an experience as possible across all apps on the device, but I know in certain circumstances it might make sense to customize the look and feel (usability, branding, etc.).