A Very Pretty Little Landaulette from William Felton's Treatise on Carriages (1796):

This is what Felton has to say about it:

A Landaulet or Demi-Landau.

This carriage has the same advantage as the landau only that the number of passengers are proportinoally less; but, for convenience, where only one carraige is kept, none exceeds it for country use.
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This was quite an expensive two-seater vehicle and a rather impressive gift on Captain Wentworth's part : Mr Felton gives the cost of a new one, fitted out with all the top level furnishings and finishes, at £156: 10 shillings and 3 pence.

It required the services of a coachman; and a footman, if he was employed by the Wentworths, could also stand on the back.

Sandy Lerner in her article in The Female Spectator Volume 4 number 1 has this to say:

This light four-wheeled conveyance gained popularity as it was well suited to England's uncertain climate in that it could be converted from an open to a closed carriage with little trouble. The landaulette was a smaller version of the landau, a very formal postillion driven veicle( postillions ride on the horses and control it from that position instead of a coachman driving from the box-seat JW) The landaulette was also known as a demi-Landau with only a rear seat. Again this is a lady's vehicle,and its inclusion denotes Captain Wentworth's extreme generosity to his wife as well as a remarkable concern for her independence

William Bridges Adams in his book English Pleasure Carriages
(1837) remarks that these vehicle ,along with their close-cousins landaus, were rather expensive to maintain in good order:

This is an expensive carriage to build and very liable to get out of order as the leather and wood work of the head is affected by cold and heat, damp and dryness. The expense of repairs is considerable