Even if one couldn't tell the leaves apart, the bark of the trees is completely different. One is smooth, the other reddish. And of course there's no way you would mistake the brown-green husk of the walnut for manchineel fruit - they are smaller in size, the husk is often cracked near maturity.
Not that we don't have issues with walnuts, either..
From what I can tell based on photographs (here walnut = Juglans regia):
Leaves: Individual leaves are very similar but walnut leaves have a pinnate arrangement, manchineel don't.
Fruit: If they're shaped like apples, or are yellowish, they're manchineel. However, manchineel are often more oval and coloured similarly to walnut (see photo on psytreasure page).
Bark: There's enough overlap here that I wouldn't rely on it.
Location: Manchineel trees are usually located on the coast or near water. Walnut trees can be found anywhere.
Range: Manchineel trees are located around the Caribbean
Walnut is Eurasian, but has North American relatives (e.g. black walnut).
Advice: If you're near the Caribbean and see what looks like a walnut tree, avoid it.
More general advice: If you're not sure what a plant is, don't eat it.
Not that we don't have issues with walnuts, either..