It depends on what I want. Going A-B-A I will use the internet. When things get more complicated I will get on the phone.
Even then it's not that simple. A few years ago I saw an advert for a fare and went to try and book it. Couldn't find it anywhere so I phoned the airline. After a bit of argy bargy I was put through to something like "special services" who were able to tell me that the fare was only available on one flight per week each way from Gatwick and not on the same day each way. With that information I was able to find availability. AN agent has access to that data immediately.
The net doesn't show you THE cheapest fare. It shows you the cheapest available fare for the parameters you input. So, here's another another twist. Suppose there are 2 fares A-B-A 100 and 200. You put in your dates and that you want 2 people to travel. If there's only 1 seat left for the 100 fare the net doesn't say that, it tells you that there are 2 200 fares available.
Then there was the time I was heading to Argentina and wanted to visit Iguassu. The agent I was dealing with pointed out that it was easier to fly to Sao Paulo and change there to fly to the Brazilian side of the falls rather than go to Buenos Aires, go from one airport to the other, and then fly BACK to the falls.
Not all agents are created equal. There are plenty who do little more than take orders in much the same way the internet does. However, find one that knows their stuff and is prepared to think on your behalf and they can be worth their weight in gold, much less their service fee.
There is one place where, unless the agent really knew me well, they probably wouldn't help. I recently went to San Francisco and deliberately came home on the A380 because I hadn't been on one. Had I gone one way to San Jose and back from San Francisco I could have had a 787 out and an A380 back but I didn't realise until after I'd booked so I went to San Francisco and back 777/A380.