Why does Tom not want to be seen in public places with Huck?

Hanging out with Huck in public places would likely get Tom into trouble. The boys in town were under strict orders not to play with Huck, as his father was a drunk, and Huck ran wild.

Shortly Tom came upon the juvenile pariah of the village, Huckleberry Finn, son of the town drunkard. Huckleberry was cordially hated and dreaded by all the mothers of the town, because he was idle and lawless and vulgar and bad—and because all their children admired him so, and delighted in his forbidden society, and wished they dared to be like him. Tom was like the rest of the respectable boys, in that he envied Huckleberry his gaudy outcast condition, and was under strict orders not to play with him. So he played with him every time he got a chance.