Nicolas Joseph Cugnot designed a steam powered vehicle in 1770. It was supposed to be a tractor that pulled artillery for the French army. It was very difficult to steer and very slow, with a top speed of less than four kilometers per hour.
The internal combustion engine was not invented until the 1880s. Steam powered vehicles were too heavy and short ranged for effective over the road vehicles in the early nineteenth century. Roads were too poor to support early steam carriages, but railroads solved these two big problems. Flanged wheels on railroads solved the steering problem and could support a heavier vehicle. A steam locomotive could pull a tender that carried more fuel and water than a road based steam carriage could. That solved the range problem or at least extended the space between fuel stops to 32 kilometers.
Source: "Cars: The definitive Visual History of the Automobile" by DK publishing(2011)

It depends on what you mean by train, since the earliest trains were horse-drawn, mainly used above and below surface for mines, and considerably pre-date the steam engine.

There were a number of early experimental vehicles in at least England, Scotland and the USA, with Cugnot's usually being given as the earliest significant one.

Trevithick constructed a steam carriage in Cornwall in 1801 (Puffing Devil). Then in 1802 he ran a commercial passenger service on a route in London with the London Steam Carriage. This seems to have been the first practical steam passenger vehicle. For comparison, the Stockton and Darlington public railway opened in 1825, initially horse-drawn. However Trevithick was already working on rail locomotives and drew 10 tons of iron and 70 men in 1804 as a demonstration.

From about 1820-40 there was a small European boom in road steam, particularly in England, and many of the technologies we currently use were invented: the Cardan shaft with universal joints, the V engine to reduce vibration, the chain drive (now mainly used by motorcycles), the Ackermann-type steering linkage, the differential and so on.

By 1840, steam carriages were on the way out in the UK. Road tolls were very high, e.g.

Liverpool-Prescot: Horse Coach 4 Shillings; Steam Carriage 48. Shillings

A Select Committee of the House of Commons in 1831 felt that the tolls were far too high, and that steam and gas propulsion should be encouraged. Not only were the steam cars faster (10-20mph, as compared with 4mph for horse) but the also did less damage to the road.

By the 1840s the railway boom was in full swing, so the opportunity for road steam was greatly reduced. It continued mainly for commercial traffic until the 1950.

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Main source for this was Steam Road Vehicles, C St. CB Davison, 1953