Worksheet and workbook are terms specific to Microsoft Excel. A workbook may consist of one or more worksheets. A spreadsheet is the same thing as a worksheet except it is a general term the can be used to describe handwritten spreadsheets as well as spreadsheets created by programs other than Excel; however, Excel worksheets are often called spreadsheets.
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The two terms essentially mean the same thing, now.
However, the older versions of spreadsheet software only had one 'sheet' in a file. I think at Microsoft Excel version 5, they added the additional sheets that could be stored in a file. So now I notice the option of saving a file as a MS Excel 4.0 Worksheet (only one sheet in a file), or a regular spreadsheet. One advantage of saving as the worksheet, it saves the current sheet that you're looking at and if importing into a database or some type of mail-merge, you don't have to choose which 'sheet' to use.
Both terms can be used for the same thing. In Excel, the spreadsheet is referred to as a worksheet. Spreadsheet is a more broad-ranging term. Excel would be referred to as an electronic spreadsheet program, as opposed to an electronic worksheet program. There are many other electronic spreadsheet programs. Excel refers to the file as a workbook and individual sheets within it as worksheets.
A worksheet and a spreadsheet are really just two names for the same thing and can be used interchangeably.