Should I fill in the spaces between pieces in a wood floor?
It's largely an aesthetic issue. Obviously the gaps will accumulate some dust and dirt, but once coated, cleaning shouldn't be that difficult as the finish will make very thin gaps disappear, and will reduce the roughness and volume of the large gaps. The solution to that is matching the wood with a filler, troweling or knifing it on, and lightly sanding the putty with a finer grit (a sanding pole helps) to smooth it and remove any excess material that gets onto the wood proper.

I recently did some puttying on a laminate floor that I had replaced sections on, and with the proper color, it can really make a difference visually, but it was a lot of work. With the laminate, it's not really possible to finish so, the putty definitely started to pick up fine dirt which actually helped blend it to the pattern on the laminate. With this floor, when you lay down the polyurethane you'll be protecting the filler so you won't have the same issue with discoloration. Honestly, I've seen a lot of old two-flats in Chicago with this and it often fits in with the general character of the home. But if this were new construction, and the walls, trim, and ceilings were flawless, it would certainly stand out.

I guess the decision you have to make is can you live looking at this every day? Is it going to be visible, or will you be throwing an area rug on it? Is it in a room that doesn't see visitors? Do you have old, brick walls, and does it fit the general character of the room? Are you OC enough to sit there working whatever square-footage you have?

SEE

Hardwood Floor Gaps &amp; Cracks
How to Fix Hardwood Floor Gaps