20 pin motherboard and 24 pin psu

Not true. You need an adapter to go from a 20 pin PSU plug to a 24 pin (often called 20+4) motherboard socket, but not the other way. Just separate the added on 4-pin from the plug-in connector. Depending on your motherboard, sometimes you don't even need to separate it; you can just push it down still attached, as long as it doesn't hit anything on the motherboard next to the 20 pin socket. However, usually there is another 12v, 4 pin CPU power connector on the motherboard that you must connect with a 12v 4-pin cable from your PSU. This is a different connector than the 4-pin that is part of the 20+4 connector, and if you haven't connected it, that would explain why your system isn't booting.

But when I plug in the old 250w psu fans are running for 30 secs, cd lights are blinking and stopping. But when I plugs the new 350w (Logisys mini atx) not even that is happening.

you should post your system components. MB, CPU, mem, vid card, etc. How old is your MB? Was it working before? Your symptoms sound like a mb or cpu failure -- the start stop is a start-up attempt that is aborted by the psu when it senses something is wrong.

Motherboard is intel. Proess: intel p4 1.6gh, videocard ati radeon 7000.

The problem started like this. Twice last time when I shut downed it itself restarted. Then I removed the plug to switch off. Next time when i turned on only the fans are running and no bios or power on.
Now the condition is like this, if I connect the old psu the fans will run and light on cd will blink for 30 secs. there is a green light on motherboard.
The difference I noted between new and old psu is, in the new one there is no wire connected to the one hole of 8th row. In old one it was there.

Sounds like a dead mb or cpu. Try resetting the bios, though with these symptoms, I doubt it will do anything. Then, disconnect everything from the mb except the psu and the on/off switch, and try again. If you get the same symptoms: Do you have another cpu you can try? Or, another mb you can try your cpu in? My money's on a dead mb. If it boots, then it's one of your peripherals, likely the hard drive.