Concrete, or visual, poetry forms a shape from the words, and the shape adds to the meaning of the poem. Shaping poetry to enhance its meaning has its first precursors in Alexandria in the third and second centuries B.C.
Nonsense poetry is a fun frolic with words that may or may not get somewhere on the road to meaning. A perfect example is Lewis Carroll's "Jabberwocky": "All mimsy were the borogoves, / And the mome raths outgrabe." Though some contend to find meaning in "mome raths outgrabe," for all intents and purposes, the words remain meaningless nonsense.
The border, then, between concrete poetry and nonsense poetry is the level of meaning that is attempted. It isn't possible to say that nonsense poet Edward Lear meant for there to be no meaning at all in
There was an Old Derry down Derry, who loved to see little folks merry;
So he made them a book, and with laughter they shook at the fun of that Derry down Derry.
though it does seem safe to say that Lear intended no deep and profound meaning to be conveyed other than that he intended to make "little folks" merry and shake with laughter.
On the other hand, concrete poets can and often do impart significant meaning through their poetry--meaning that is enhanced by the shape--as is the case with Court Smith's concrete poem "Dove's Song," the words of which are:
Cast away your arms --
dress for peace
wrap in nature's cloak --