Generally, you would need to file if you have income in the UK that doesn't get taken out automatically - that is, if you only get money from a salary (with tax taken out before you get paid) or a UK bank account (they take out tax on interest before you get the interest), you don't have to file.
You would need to file if you have income coming from another country.
You need to file if you are self-employed.
If HMRC thinks you have to file, they will usually send you a letter (but I wouldn't wait for the letter if I already knew that I had to file.)
Phatbeetle, if you are already getting tax taken out by PAYE and you don't have any other income, and they haven't messed up the PAYE if you do self-assessment you are probably only going to have a minimum amount of tax to pay or to be credited, if anything - a few pounds or pennies.
I'm not sure why you were told that you need to file self-assessment if you do everything by PAYE. Are there other issues?
Generally, if your tax is straightforward and you don't have complicated issues, I find that UK Self Assessment is much easier than a US 1040.
I don't file online, but I think the advantage is that if you file online, the deadline is longer.
ETA: I'm not a tax expert. Just speaking from personal experience.