In very basic terms (there are plenty of complications and exceptions), a DSLR has an image sensor that dwarfs that of your average P&S. However, compared to a P&S, DSLR sensors have far fewer pixels relative to their surface area i.e. a lower pixel density. While this might sound like a disadvantage with all the marketing suggesting more pixels = better, a lower pixel density can be beneficial. If you take a 12MP DSLR and a 12MP P&S, the DSLR will have larger pixels, potentially allowing for better light gathering than the P&S. Along with the lower pixel density comes a reduced likelihood of image noise from less electronic interference between pixels and not having to ramp up light sensitivity as much with DSLRs, particularly in low light environments.
With the dragonfly (forgive my entomology) and the flower, what you're seeing is the effect of having a shallow depth of field, which refers to the distance between the nearest and farthest objects that stay in focus. The chances are that a lens with a wide aperture, also known as a fast lens (because you can shoot at a faster shutter speed than a standard lens but allow the same amount of light in due to the wider aperture) was used to take both shots. As most P&S cameras have limited manual aperture control, which sets your depth of field, it can be tricky trying to achieve that effect with them.
DSLRs: Canon EOS 7D Mark II, Canon EOS 70D
CSC: Canon EOS M3
Lenses: Canon EF-S 15-85mm f/3.5-5.6 IS USM, EF 50mm f/1.8 II, EF 100-400mm f/4.5-5.6L IS USM, Canon EF-M 18-55mm f/3.5-5.6 IS STM, Canon EF-M 22mm f/2.0 STM