First, what they have in common. Both books were written by would-be authors attempting to gain a reputation, so I guess they tried travel writing as a convenient starting point. But there the similarity ends. Mrs Trollope (mother of the famous novelist) was something of a pioneer. Few Europeans had visited the southern States at this time and written about their experiences, and so Mrs Trollope's account would have the value of a writer looking at unfamiliar surroundings. In contrast, Mark Twain's account is of a cruise from New York to the Mediterranean, and the places he visits (at least as far as I have read) are quite familiar: the experience of travelling by ship, then visiting Gibraltar and Morocco. So is that the difference between the two books? Not quite.
Perhaps the most noticeable difference is the author's (implied) position. Mrs Trollope may have started to write a travel book to make money, but she clearly got caught up in the surroundings; she responded with a vigour and commitment that is compelling. When she complains about American males spitting in the street, you feel that this is not a casual observation but a fundamental objection to the American way of life.
I hardly know any annoyance so deeply repugnant to English feelings, as the incessant, remorseless spitting of Americans. [Chapter II]
Compare this to Twain's description of the cruise ship's company viewing the rock of Gibraltar:
Here in Gibraltar he corners these educated British officers and badgers them with braggadocio about America and all the wonders she can perform. He told one of them a couple of our gunboats could come here and knock Gibraltar into the Mediterranean Sea![Ch VII]
Cleverly, Twain is able by fictionalising his comment to disclaim responsibility for the statement. But in so doing, he misses the opportunity to state his own ideas; the narrative describes various travellers on the voyage with Twain. These characters might be fictional, but their conversation is reported as if heard. Unlike Mrs Trollope, Twain does not make his own position clear, and he as a result runs the risk of being labelled as belonging to the group of philistine American tourists he mocks. In fact he describes himself as a participant, being swindled by a local trader when he goes to buy some gloves. His companions subsequently tease him for letting himself be defrauded:
Every now and then, my glove purchase in Gibraltar last night intrudes itself upon me. .. They let me alone, then, for the time being... But they had bought gloves, too, as I did. We threw all the purchases away this morning. [Ch VII]
In other words, while Mrs Trollope places herself firmly outside the world she is describing - she is English, and not American - Twain depicts himself as a member of the group. By placing himself on the inside of the group, I guess he makes the mockery more gentle for his readers, who will no doubt be Americans (I'm one of you, guys, so I'm not really being spiteful). But you can't both criticise your fellow Americans and at the same time be one of the targets you mock. I'm afraid to say that from this book I am tempted to regard Mr Twain as rather stupid, certainly no cleverer than those co-travellers he mocks, while Mrs Trollope, who states quite clearly her limitations as a writer (no attempt to write about political systems, for example) comes across as a far more serious and committed commentator. Strangely enough Mark Twain recognises her achievement in his Life on the Mississippi:
Of all those tourists I like Dame Trollope best. ... She lived three years in this civilisation of ours; in the body of it - not on the surface of it, as was the case with most of the foreign tourists of her day.... Mrs Trollope, along of them all, dealt what the gamblers call a strictly "square game". She did not gild us; and neither did she whitewash us. [suppressed passage from Life on the Mississippi, quoted by Donald Smalley in his introduction to Domestic Manners]