The biggest advantage is definitely the instant connection to information. Every option available comes up at the click of the mouse, and you can customize your searches. The quicker one can get information, the faster one can analyze it, instead of wasting valuable time searching.
To me, the best part is that it does not take clutter, and information can be bookmarked for future and repetitive usage. Also, printed documents are more likely to be outdated than a constantly updated online site. To me, it is the best way to "go green".
The previous post is spot-on concerning the changes in research during the past 30 years (or less). In the past, a researcher would have to camp out in a library (or libraries) to peruse the many sources that may or may not be available to him. Then, a great deal of time would have to be spent skimming throught the texts in the hope of finding a few shreds of needed information. Today, you simply go to Google, type in a few key words, and voila!--the sources are literally at your fingertips. Since most research material has now been referenced online, the ease of attaining it has become a simple process.
Are you asking about the advantage of online search as opposed to looking things up in actual physical books? If so, it's like the advantage of telephoning someone who lives 10 miles away over walking to talk to them.
As someone who had to look up lots of stuff in the days before the internet, I tell you that it was no fun at all. For little things, you had to have a set of encyclopedias. That cost a lot of money, quickly went out of date, and gave you access to only one perspective.
Then when it came to research papers, things were even worse. You would have to find out where the information was located and go there and get it. For example, I remember having to drive 50 miles to the state archives to find some information that I could now get standing here in my kitchen.