What's the difference between a sized diaper and a One-size diaper?

The main difference  is sized diapers are made to fit a specific weight range.  Typically these diapers come in newborn or XS, small, medium, large, and extra large.  Since they are made for a smaller weight range, they are less bulky and sometimes easier to get a good fit on a baby.  They have nothing to adjust the height of the diaper, it is the size it is, if that makes sense.

One-size (OS) diapers are made to fit a large weight range, usually 10-35 pounds.  These diapers are bulkier on a smaller baby, and fit more trim as baby gets older and bigger.  They do not generally fit a newborn well.  Sized XS or Newborn diapers are best for tiny babies.  Even though a lot of OS diapers state they'll fit around 8lbs, most people find they fit best once baby bulks up a bit around 10 pounds.  OS diapers feature either a snap down rise or a fold over rise.  A snap down rise features a couple rows of snaps on the front of the diaper, you can snap these to eachother to customize the rise/height of the diaper.  A foldover rise has one row of snaps on the outside of the diaper, and another row on the inside.  You fold the inside row down and facing out of the diaper, and it makes the rise/height of the diaper shorter.

Left is a size Small Bumgenius.  Right is a OS BumGenius

The snaps on a OS Bumgenius

The small next to the OS snapped down to the smallest setting.

The difference in bulkiness in the small and the OS on the smallest setting.

Here is the small on Claire at about 13 pounds.

Here is the OS on Claire on the smallest setting.

Here is the OS on the medium setting.

And here is the OS all the way unsnapped to make it a Large.  You can see how big it is right now!

This is a different brand of OS diaper - a Rumparoo.  This has 4 size settings.  This is on XS (it's a little too short for us now) The Rumparoo is one of the few diapers that really does fit a smaller baby (under 10lbs) - but it doesn't quite last as long because it's smaller than other OS diapers.

This is the small setting, fits perfect now.

The medium setting.

The largest setting.  Still lots of growing room, but not as much as the BumGenius.