One way damselflies differ from dragonflies is in the head shape. From above a damselfly's head is relatively short (from front to back) and wide with the eyes capping the ends—what could be described as “hammer-headed”. Unlike dragonflies, there is always a gap between the eyes which is at least the width of one of the eyes—usually more. See the image at right which illustrates the distinctive head shape.
Does this hammer-headed condition give damselflies some sort of advantage? I think it does. Damselflies frequently perch on long, skinny, near-vertical things like grass stems, sedges, and twigs, with the head facing whatever they’re grasping and the abdomen pointing away. The widely separated eyes provide them an unobstructed view around narrow perches, which means they can keep an eye out—well, both eyes out—for whatever they need to look out for. I would guess a narrow perch does not block the damselfly's view at all even though it's right in front of the face.
The photos below illustrate this point. This California Spreadwing (Archilestes californica) was perching on a sedge at the Cottonwood Recreation Area on the John Day River in Oregon. After a few profile photos, he stayed put while I angled around for the front shot. Very cooperative!
Two views of a male California Spreadwing (Archilestes californica) perching on a sedge stem.
What do they need to look out for? Predators are an obvious choice, although I’m not sure how noticeable damselflies are to predators while perching. It must happen sometimes, though. They do often perch-hunt however, waiting for prey to fly within striking range and then taking their lunch back to a perch to finish up. Mature males are also always on the lookout for females, so whatever enhances their view is good for that too.