Yes, from many places in the south west it's both cheaper and more convenient.
There is a coach from Reading station to Heathrow. If you search the national rail planner for a journey from somewhere in the west (e.g. Bristol) to LHR it will generally recommend this as being both quicker and cheaper than taking the train in to London and back. A taxi would presumably be quicker than either as you don't have to wait for the coach's schedule and it doesn't have to stop at other terminals. In terms of price it depends how many people are travelling and how much luggage you have, a quick search for taxis from Reading suggests £33 to £55 for a taxi each way. You can get the train and coach ticket together or book them separately, the coach ticket price is apparently £19 single, but you can book an early-bird family return for £35.
Depending on where you are in the south west, going via Woking might be better instead; there is a very similar rail air coach.