In previous times, web browsers have consistently aimed to restrict access to the operating system and browser chrome. The main concern revolves around trust and deception - should a website be allowed to suddenly start opening dialog boxes or accessing files on your hard drive just because you visited it? There's also the danger of falling victim to phishing attacks, where a seemingly legitimate URL actually leads you to a malicious site designed to deceive you.
The direction things are heading in is quite nuanced. While modern browsers now support GPU usage and offline storage through HTML5, security measures like sandboxing are always being carefully considered. On the other hand, there are instances where browsers are becoming more strict, such as with restrictions on cookies when using the file:// URI.
Many applications nowadays are transitioning into web apps, but the process isn't as straightforward as simply navigating to the app in your browser. They could be mobile web widgets, or apps acquired from platforms like the Palm Pre's app store, where most apps are essentially web-based. The level of trust varies depending on the situation; for instance, I might feel safer granting certain OS permissions to an app from a reputable store that has been thoroughly inspected and signed by the developer, as opposed to giving those same permissions to a random website I stumbled upon.