How do you connect these to your computer? Which one would be easier to burn?
The information you seek can be found by comparing the data sheets: AT89C51, AT89S51, however you will see in this first link that it states:
Not recommended for new designs. Use AT89S51.
So that probably makes your last question irrelevant, you should use the AT89S51 regardless.
With regard to your second question (which I interpret as "connecting to a development host"), I suggest that you use an off-the-shelf development board unless you have the means and skill to produce your own. In-system programming can be achieved through the UART (there's an app. note in the link above), so you'll need a PC with an RS-232 serial interface and a NULL-Modem cable (the latter may come with the board). If you do not have a serial port on your PC, get a USB serial port adapter.
Obviously you need the development tools; at leas a compiler, and for debugging you'd benefit from an emulator; again the link above has references to tools.
Overall; read the manufacturer's documentation is the answer to all your questions! ;-)