Nature has some very unique properties, and biomimetics is the attempt to copy them. Here are some examples:
The lotus leaf's special power is the ability to stay clean, despite living in muddy water. From this talent, it has become a symbol of purity.
Water harvesting (Namibian stenocara) beetle
This beetle has patterns of water attracting and repelling surfaces on its back to catch fog and drink it.
The gecko's feet can stick to almost any surface, yet don't get dirty!
This can change the hardness of its body.
This is possibly the most famous, spider silk can absorb the most energy before breaking. It's quite impressive. Imagine a fibre strong enough to stop an insect in mid-flight, yet thin enough to be nearly invisible. Spiders can make this effortlessly, while to make this we need extreme control over temperature and pressure - and we haven't succeeded! Sadly spiders are cannibalistic, which rules out spider farms.
The colour of some butterfly wings comes not from pigments, but from destructive interference of reflected light, a bit like colour from a CD. Not sure what the advantage of this is over pigments though.
The dragonfly has amazing flight capabilities. Its body encounters up to 30Gs of force. Combat pilots require bulky complicated systems to help them survive the G forces their planes can easily apply. Dragonfly bodies have a liquid cushioning system, where the liquid flows to the side with the most G forces, and protects them!
Resilin, a protein found in insects, is very efficient at storing elastic energy and very resilient (can bend many times).