Start with the honest answer
If you are one person, travelling in the middle of the day, going somewhere near Paddington, and you booked three weeks ago, take the train. It is roughly 1 hour 40 minutes on a fast service against about 2 hours 30 minutes by road, and an advance single is cheap. We are not going to pretend otherwise.
The interesting question is the other cases, and there are more of them than you would think.
Door to door, not station to station
The 1 hour 40 figure is Temple Meads to Paddington. Your journey is neither of those places. Add getting to Temple Meads and parking or paying for a taxi at that end, then add the Underground or a London taxi at the other, plus the walking with luggage in between.
For a central destination that adds perhaps 45 minutes. For Canary Wharf, Wimbledon, Croydon or anywhere in the outer boroughs it can add well over an hour, and at that point the two options are level on time and the car has been more comfortable throughout.
The arithmetic for two or more
Rail fares are per person. A private hire fare is per vehicle. That single difference decides most of these comparisons.
One person: the train, usually. Two people: close, and it depends how far in advance you booked. A family of four or a team of six: the car, comfortably, and with no changes and no reserved seats in different carriages. Eight of you in a minibus is not really a contest at all.
Luggage and awkward items
Great Western trains have limited luggage space and it fills. Four large suitcases, a cot, golf clubs, a cello, an exhibition stand, a fortnight's kit for two children — all of it goes in a boot without discussion, and none of it is fun on a crowded 07:30.
If you are carrying anything you would not want to lift twice, the calculation has already been made for you.
Times of day when there is no train
Before about six in the morning and after about eleven at night the comparison stops existing. Add Sunday engineering work, which regularly replaces part of the route with a bus, and strike days, and Christmas, when the railway shuts entirely and the roads are the emptiest they get all year.
A 04:00 departure for an early Heathrow flight is a car booking or it is nothing.
Working, sleeping and arriving
Some people work better on a train, with a table and no responsibility. Others get more done in the back of a quiet executive car with no announcements, no person opposite and no need to change at Reading. Neither is wrong; they are different products.
What a car does guarantee is one seat for the whole journey, which on a busy Friday afternoon out of Paddington a reservation does not always achieve.
A summary you can act on
Train: one person, daytime, central London, booked ahead, travelling light. Car: two or more people, real luggage, an outer-London address, early or late, a same-day return for a meeting, a group, or a day when the railway is not itself.
If you want to test it, get a fixed quote for your actual journey and compare it against the total door-to-door cost of the train rather than the headline fare. That is the honest comparison, and it is the one we are happy to lose sometimes.