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 --
I don't think a great concrete poem is nonsensical in any way. Nonsense is, well, not making any sense at all or just rambling without a point.
A good concrete poem is in the shape of what it is discussing. Hence, a poem about Christianity could be in the shape of a cross, or a poem about rain could be in the shape of a raindrop. On top of that, a moral or message of some sort is offered in any good poem. Concrete poems often offer a sensory experience, in the same way as a haiku.
I like a good concrete poem because it offers a visual representation along with the ideas. It's not the most sophistocated and high-brow form of poetry, but that's okay. Younger students usually enjoy them and like to try to write them. Anything that inspires students gets a thumbs up from me.