A lady licking an envelope. An intimate thing. It might be only the bill from the coal-man she’s paying, but it has the feel of something else: an assignation, a confession, an apology, a breaking-off. Would this woman in her deep-blue day dress and jacket be so unguarded if the artist had been a man? Mary Cassatt (1844–1926) was a femme sérieuse who painted women of quick wits and tender instincts. No grubby models, no ballet rats, no laundresses, no absinthe. Her sitters, you feel, would write a thank-you note, send flowers, recommend a dressmaker.

Mary Cassatt: An American Impressionist in Paris, at the Musée Jacquemart-André, is the first French exhibition to be dedicated to the Pittsburgh-born painter since her death. Cassatt moved to Paris from Philadelphia in 1865. She was an outsider twice over: a woman and an American. Still, she had taste, money, family, ambition, an apartment near the Place Pigalle and, after the success of the impressionist exhibitions, a château outside Paris. ‘What’s she got that we don’t?’ Camille Pissarro sulkily asked his son after visiting ‘Mlle Cassatt’ in April 1891. ‘Money, yes, just a bit of money.’

She was very much ‘Mlle Cassatt’. She never married, and after her sister Lydia died, never found another companion. But there is no one to beat her as a painter of mothers and children. Her subject was the ‘modern Madonna’. Instead of well-behaved Christ-childs waving benedictions, Cassatt’s babies squirm, wriggle, fidget, peel off their socks and blow burpy little raspberries. Her children have weight. She catches mothers shifting position because their cradling arm has gone dead.

Cassatt excels, too, at the shifting relationships between friends when one has a baby. A Japanese-inspired drypoint shows two women on an omnibus, one fussing and readjusting her bonneted daughter, utterly absorbed, the other looking out of the window. ‘Didn’t we used to talk?’ she might be thinking. These etchings are the exhibition’s highlight. Cassatt takes the language of Hokusai and Hiroshige and translates it into the French spoken in Paris salons and the dressing rooms of rich Americans abroad. Another drypoint, ‘The Woman Bathing’ (1890–91), is sensual and private. A woman, her back to us, washes her face, chemise rolled down to her waist. Compare her undressed ease in Cassatt’s company with the bathers painted by Cassatt’s friend Edgar Degas. His women feel ambushed and invaded.

The critic Joris-Karl Huysmans said: of course Cassatt’s good at children. She’s a lady; it comes naturally. ‘Only the woman can sit a child, dress them, put in the pins without pricking her finger,’ he wrote, as if Cassatt’s genius were merely a matter of getting the nappy pins right.

At her best, Cassatt paints the slumpingly tomboyish ‘Little Girl in a Blue Armchair’ (1878), forced into frills and tartan socks for a portrait and determined to hate every second of it. At her worst, there is a touch of the Shirley Temples to her poodle children in silly hats.

Still, you realise just how good she is when you compare Cassatt’s pastel ‘Sleepy Thomas Sucking His Thumb’ (c.1884) with Camille Corot’s weird, wooden infants with staring Stephen King eyes. Cassatt’s toddlers are sweet, dimpled, talcum-powdered; Corot’s are little poupée-people with hinged elbows and clenched hands. ‘Corot did not particularly excel at portraiture,’ says the catalogue to Corot: The Painter and His Models at the Musée Marmottan Monet. Hardly an enticement. Really he was a landscape painter: a Claude, but sludgier, a Poussin, but gloomier, a Salvator Rosa, with fewer ragged rocks. He was a pick-and-choose classicist. A hermit here, a naiad there. A sibyl, a nymph, a picturesque peasant. His bacchanals never look much fun. ‘Sigh no more, ladies, sigh no more,’ you think as his forlorn muses slouch around Arcadia. What happened to blithe and bonny? His rather gormless farm girls fall asleep over books they can’t read, while ‘Virgil’s Muse’ (1845) might be a bored first-year doing amo, amas, amat on a hot afternoon.

The loosening of his brushwork in later works looks towards impressionism; Berthe Morisot was his pupil. The painterly ‘Woman with the Yellow Sleeve’ (c.1870), bought by Lucian Freud and now in the National Gallery in London, is a virtuoso performance. The sleeve, the ribbon, the bows are done in a thick impasto of streaky-bacon oil paint. You can see why Freud liked it.

After early disappointments, the last room is stupendous. Here, Corot’s masters are the 17th-century Dutch painters Vermeer and Gerard ter Borch. In the two versions of ‘The Artist’s Studio’ (Washington, c.1868; Louvre, c.1873) Corot paints with a diffuse light and haze that is unlike anything that has gone before. His ‘Lady in Blue’ (1874), modelled by Emma Dobigny, a model from Montmartre, is rightly the poster girl for the exhibition. She is a column of colour, a shot of ultra-blue in Corot’s subdued studio. She is captivating, but solemn. Corot makes his women melancholy, as if a little learning were a burdensome thing. Cassatt’s women wear their intelligence lightly, like a particularly pretty hat from Reboux. Cassatt’s sitters drink tea from china cups and go home to husbands and babies who stretch out greeting arms. Corot’s girls, once they have stopped being ‘Clio, Muse of History’, go god knows where. What is the difference? ‘Money, yes, just a bit of money.’