Explain how Huck Finn does not mature during his journey down the Mississippi.

It might be useful to think of Huck's maturation in a less binary way; it isn't perhaps a question of whether or not he matures, but in what ways has he not yet grown as the novel concludes.
Perhaps the strongest evidence that Huck has not fully matured is found in the chapters at the Phelps farm. As Huck and Jim travel south, the many episodes they experience together show a gradual shift in Huck's thinking about race. He begins to unlearn all that society has taught him about slaves and people of color as his relationship with Jim develops. This happens when Huck is away from the racist society that sees Jim as a commodity, not a human being. However, once Tom Sawyer reenters the narrative, Huck's burgeoning moral conscience is overpowered by the boy he sees as his social and intellectual superior. Huck lacks the confidence and wherewithal to stand up for Jim and put Tom Sawyer's cruel antics to an end. Huck's dilemma is that he has changed internally, but the society that surrounds him has not changed. The piece that is missing from Huck's maturity is that he does not feel empowered to challenge a racist society. In the end, he opts to flee what he knows he cannot change—or endure—on his own.