From what I have observed I think there is only one, at times two, subtle differences. Milk being one and at times the quantity of sugar although the lines have blurred with different versions and many cooks.

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Kesari typically is made from Sooji (or Rava/coarse semolina) with water, ghee and sugar playing predominant roles followed by aromatics.

Sheera, is usually made from Coarse semolina (just like kesari) with milk, ghee, water and sugar with the emphasis more on the addition of milk in this case. From what I have personally observed Sheera tends to be less sweeter than Kesari.

But families have blurred the lines between them, I think. Whatever may be the differences, both of them are among the best Indian desserts :)