Can these words be used interchangeably when referring to a point in the world?
A good way to think of it is that a position is more defined than a place: given the chess board example, the position a piece can be in is defined by the 64 squares on the board. While you could say the knight piece is in the place of A-4, it'd be more definitive to say it is in the position of A-4, to imply that there is structure to where it can be.
On the other hand, a place is more ambiguous and requires no structure.
Position often refers to a specific location for which there are geographic or other coordinates. For example, the use "GPS Position" refers to a specific location on the earth, identified by by a latitude, longitude, and altitude (and various error estimates, called the dilution of precision coefficients). GPS stands for Global Positioning System, and in this sense "positioning" means the finding of positions for users of the system. One would be hard pressed to substitute the word "place" in either context. The neologism "geolocation" is often used by technologists and navigation specialists, but often position or "position fix" is used by the broader community.
"place" and "position" could be used interchangebly in some specific case, one that i can think of right now is related to the pieces in a chess game, you could say that the white queen is in A-3 like this - "White queen's place is A-3", or "White queen's position is A-3"
The difference though is that "position" in many cases involves a meaning of "orientation", like... "i'm in the middle of the room, facing the window", so "place" would be "the middle of the room" and "position" would be "facing the window". Position could also mean the state of your body like "laying on the floor", "sitting", "upright", etc.
"Position"could also have a hierarchic meaning, like "subordinate" or "boss", can be considered positions.
"Place" may indicate a similar situation, as in "I'm in no place of giving orders" (considering that I'm a subordinate in some case).
So, "place" and "position" have similarities, but also specific values.