A public place is a place where anyone can go: a train station, supermarket, park, street, etc.
Public space is a bit harder to define. Mostly I think of public space as opposed to private space.
a. In Chicago, Northwestern Station is a building for boarding commuter trains. It is a public place. Smoking is not allowed inside the station.
b. Inside the station is a glassed-in room for smoking. I suppose that could be called a public space.
c. Each person and each culture has what is known as private space- the space inside which one does not feel comfortable allowing strangers. Westerners tend to feel crowded more easily than Asians do. It happens to me sometimes when I am waiting to board a bus. I allow what I think is a bit of space between me and the person in front of me, but often another person will step into that space. I may consider that action a bit rude, but the Chinese person who stepped ahead of me may simply think I'm a bit stupid for not moving up. With that understanding of private space, anything outside of that might be considered as public space.
d. I live in one building of an apartment complex. To reach my apartment (a private space) one must first pass a gate to get inside the complex. Anyone who lives in any of the buildings can reach this public space, but not anyone at all can wander in off the street. Also, the spaces inside the building, but not inside any particular apartment, can be considered public space: entryway, stairs, hallway, mail room, laundry room, etc.