Most people search for a round-trip because that's what the form asks for first. Fair enough. But it's worth spending two minutes checking the other way too.
Domestic trips: sometimes splitting is cheaper
Inside the US, airlines often price each direction on its own. So a round-trip is sometimes just two one-way fares added together, and sometimes it isn't. Every so often, two one-ways come out cheaper, especially when the outbound day and the return day sit at very different price levels.
There's another perk. With two one-ways you can fly out on one airline and come back on another. If Delta has the better fare going and someone else has it coming home, you're not stuck with one airline for both.
International trips usually go the other way
For flights abroad, a round-trip is very often the better price, and a one-way can cost surprisingly more. Some countries also want to see proof that you're leaving again, like a return or onward ticket, before an airline will let you board. If you're going one-way to another country, check the entry rules for your destination first.
The catch with two separate tickets
Two one-ways are two separate bookings. If you need to change one, the other doesn't move with it. If your first flight is canceled, the airline you're flying home with isn't going to rearrange anything for you. That doesn't matter much on a simple trip, but it's a real difference.
A quick way to check
Price the round-trip. Then price each direction on its own, on the same days. Compare the totals, including bags and seat fees. The airport can change the answer too, so try nearby ones if you're flexible.
If that sounds like a lot of browser tabs, send us your dates and we'll compare them for you.