Replace the word "apps" with "websites" in your post and it is just as true. Especially if you use your browser for everything and state for everything you do is contained in your browser....
Not saying "don't run the browser version, apps are better" - merely that if you think you are getting away from that sort of thing by running the web version.... you aren't.
Running the web version can give you better battery life though because safari will be killed in the background and a lot of apps can/will run in the background by default.
Granted, there are no guarantees either way.
Using a browser can reveal PII as well, but at least with them, there is some measure of control over what it stores, connects to, and permissions guiding what it has access to. Their processes can be killed, and not operate in the background, or in a completely opaque manner like an app.
To take one example:
If I want to know the weather forecast, and visit a site to obtain one, it's implicit that I'm telling the company that is where I'm located, or plan to be, or have it roughly geo-located according to IP. In exchange, it shows me an ad along with the information, and that's pretty much the extent of it.
If I use the company's app, it will ask me to permit Location Services to determine precisely where I am, and ask me to agree to a EULA to proceed. That legalese (which people never read) may, but may not contain the fact that it will collect other information unrelated to the primary purpose of the app, share or sell that information with third parties, for purposes unrelated to the simple need to provide the relevant forecast.
If that app is allowed to operate in the background, it can collect and dispatch that information unabated, even when I haven't asked to see the weather for a particular place; it will know about it, and forward the information. The Weather Channel and WeatherBug are two that have been found to do such things.
The financial incentives are just too great for developers, large and small, to ignore.
Because so many tech companies live by the creed of "It is easier to ask forgiveness than it is to get permission," they do not get the benefit of the doubt, at least from me.
Scott McNealy is famously quoted as saying there is no privacy on the internet, and the a large extent that's true. But, I'm not going to just roll over, and play dead either.