In the (quantity, income) space, not being everywhere differentiable would mean that we would observe gaps, discontinuous jumps, or kinks, in the corresponding graph of the Marshallian individual demand function.
GAPS: if there was a gap, it would mean that for a certain range of income, the demand function does not exist. In other words, it is not defined for a certain range of income (but it still is defined with respect to prices). This could be interpreted that for this range, income does not affect quantity demanded. So "differentiable everywhere" excludes such a phenomenon.
DISCONTINUOUS JUMPS: If there was a discontinuous jump, it would mean that a specific level of income induces a "threshold effect" on quantity demanded. So "differentiable everywhere" necessitates that no such effects exist on quantity demanded.
KINKS: If there was a kink, we would have a "second-order" threshold effect, where quantity demanded would change at a discontinuous rate with respect to income. So "differentiable everywhere" excludes such phenomena.
These should now be "mathematicized" and projected back to the properties of the utility function (if, that is, they are relevant to it).
Note: In many utility maximization problems, income does not enter the utility function, but it is part of the problem's constraints. In others, we postulate a utility function as a bivariate function of one good and income.