If you're travelling from Heathrow into Central London or across London then opt for the Piccadilly Line as it's often quicker than the Heathrow Express, and cheaper too!
Updates
London Zone 1 to Heathrow (Zone 6) can be as cheap as £2.20 single if you have an Oyster card and travel off-peak between 0930 and 1600 or 1900 onwards (and anytime weekends). Otherwise £3.80, or £4.00 if paying in cash. A huge difference from the exorbitant £16.50 (£19.50 if purchased on the train) that Heathrow Express charges.
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