Usually the pieces shown on the table are removed as you use them. I don't usually pay much attention to them unless I know I am running short and want to see how many I have left. I have not noticed any discrepancy in what is left on the table and what I have available to play - but there are a lot of graphic refresh issues since the release of version 2.
For most scenarios you will have 5 settlements, 4 cities, 15 roads and (for Seafarers scenarios) 15 ships. There are only a couple scenarios (Grand Catan & Oases) that use different numbers of pieces. For Cities and Knights, you have 2 of each strength knight (basic, strong, and mighty).