Crazy Problem

I have a 1999 S-10 Blazer with a 4.3L Vortec. Why is it that it doesn't start when I turn it over without jumper cables hooked up to the battery, but it starts up when I jump it with another vehicle? The Blazer turns over fine without jumping it. The battery isn't dead at all. I even bought a battery with higher cranking amps and it still doesn't start without jumping it.
Has anybody came across this problem or know what would be causing it?

Appreciate any help!

I would check and clean the connections. terminal to the battery, cable to the terminal, cable to the starter and wires at the starter. On the connections at the starter, check for loose or broken wires. mabey a wire that is hooked up but has a green corrosion look to it. same with the cable that is formed to the terminal.
when I say clean I mean remove and scrap or use emery cloth till shinny

Terminals are spotless and I cleaned the other connections with a fine grit sand paper so that they are shiny and I still have the same problem.
Any other ideas?

Check and make sure your charging system is charging properly, and especially that your battery is good and not weak, or need charging.

Assuming your battery is good, connections are strong, check your fuel pressure before(without jumper cables) and then after(with jumper cables). If the minimum pressure is not reached without cables and is after you put the cables on, you have found the no-start. In this case, check connections at the fuel pump itself and make sure they are not corroded. If connections are good, fuel pump is recieving the necessary volatge, I would then at this point be looking at the fuel pump. Before all of this though, you have to make sure charging system is up to snuff, and proper fuel pressure is reached.