In most cases emulators are best to test any website or application.
You can select n number of devices or resolutions through the web because now a days browsers have inbuilt resolution wise or device base testing feature which is easy to use and less expensive for the organization. Moreover, designers or developer knows the tactics of how to write proper CSS or script to make feature which can work well on all the devices. However, sometimes it's hard to identify the problem in emulator but in most cases it's very useful.
For any mobile application emulators are everything. Mostly developer prefere to run or test their application on emulators as those are available on wide range and easy to use. It can reduce the cost to company. Easy & fast way to test. However, sometimes real device are required to test application which has feature to scan through camera.
All in all, emulators are very handy to test anything you want.