There are several key differences here.

First, the Olympus lens needs to cover a smaller sensor — Micro Four Thirds uses a sensor which measures 17.3×13mm, while the Nikon DX camera uses a 23.5×15.6mm sensor. That means the Olympus lens needs to project an image circle with a diameter covering 21.6mm, while the Nikon lens needs to cover 28.2mm. That means everything in the Olympus lens can be correspondingly smaller.

(This sensor size difference is also why the Olympus lens has a shorter focal length, but approximately the same field of view.)

Additionally, the mirrorless design of the Olympus system allows flange focal distance ­— the distance from the lens mount to the sensor — to be smaller. That's about 19mm for Micro Four Thirds, and 46.5mm for Nikon.

Third, the Nikon lens features vibration reduction, while Olympus relies on that feature in the camera body.

I think both lenses basically are the "smallest possible lens", given the circumstances. However, it's also worth noting that larger-than-smallest-possible also can have ergonomic advantages: there's more room for focus and aperture rings, and those rings can have more travel.

There are also various other lens design choices which can influence the diameter — see more about that at How does the lens diameter influence photo quality? — but as a general rule there's nothing there that's going to be a big deal for a kit zoom.