TL;DR - As of Feb 2017, Progressive Web Apps are a sufficiently powerful platform that Twitter has moved all of their mobile web traffic to a React PWA.

As of August 2016, Progressive Web Apps actually offer more hardware access than commonly thought. Here's a screenshot of whatwebcando.today from my Chrome 52 stable on Android:

Hardware access includes

Upcoming hardware access

These features are being implemented or already work in some browsers:

Another important point to note is that the Origin Trials Framework (implemented in Chrome) enables manufacturers to expose and test hardware (or software) capabilities without having to go through the standardization process. For example, a phone maker could expose an API for reading the values of a pressure sensor, refine it, then submit it for consideration to the W3C.

Besides hardware access, there are also software features traditionally employed by native apps that are now available to web apps.

Traditionally native features that PWAs can also use

These features cover a lot of use cases, and many popular native apps nowadays could be rewritten as PWAs. Take Slack, for example. Its open source alternative, Rocket.Chat, is building a PWA version. For more PWA demos, see https://pwa.rocks.

Native-like features coming to PWAs

handling intents — for example, sharing a page to another app, or being the share target, e.g. a PWA chat app that receives an image to set as the user’s avatar

Native Android features not yet available to PWAs

access to the fingerprint sensor (under development)

contacts, calendar and browser bookmarks access (lack of access to these could be viewed as a feature by privacy-conscious users)

alarms

telephony features - intercept SMSes or calls, send SMS/MMS, get the user's phone number, read voice mail, make phone calls without the Dialer dialog

low-level access to some hardware features and sensors: flashlight, atmospheric pressure sensor

system access: task management, modifying system settings, logs

Progressive Web Apps offer features that native apps lack

Final note: PWAs run, with the same codebase, on the desktop as well as most mobile devices. On desktop environments (ChromeOS, and later Mac and Windows), they're launched in the same way as other apps, and run in a regular app window (no browser tab).