Just because you can use a certain web font, it does not mean that it’s a suitable choice. Aesthetic reasons aside, not all fonts look as good on screens as others. As with any medium, you should develop an eye for the distinctive characteristics of different screens, and understand how well different typefaces behave with each.

Not only are digital displays different from the printed page, but there are many kinds of screens that may display web fonts. Let’s compare:

typefaces

Scalable outlines of Aptifer Slab

A screen grab of 1-em text in Mac Safari, enlarged to show the rendering

A screen grab of 1-em text in Mac Safari, enlarged to show the rendering

1-em text on the LCD screen of a Macbook Air

1-em text on the LCD screen of a Macbook Air

1-em text on the LCD screen of a Nokia Lumia 920

1-em text on the LCD screen of a Nokia Lumia 920

1-em text on Kindle E-Ink screen

1-em text on Kindle E-Ink screen

You should try and have a look at these pages in a variety of web browsers (Internet Explorer, Firefox, Safari, etc.) on a variety of platforms (Windows, Mac OS, iOS, Android). That way you can see exactly how differently the same material can look from one environment to another.