A great difference in the transition from varieties to schemes is the presence of non-reducedness. Sometimes on a given scheme there are different natural scheme structures and with respect to one, the scheme is non-reduced, while with respect to the other one it is reduced.

An example is the scheme of nilpotent $n \times n$ matrices for $n \geq 2$. One way to give a scheme structure to the set of nilpotent matrices is to observe that an $n \times n$ matrix $X$ is nilpotent if and only if $X^n$ is the zero matrix. Thus all the entries of $X^n$ are polynomials (homogeneous of degree $n$) in the entries of $X$, whose vanishing set is the set of nilpotent matrices. As has been stated in the question and noted in a comment by Victor, the trace of a nilpotent matrix is a function that is identically zero on the set of nilpotent matrices, but it is not contained in the ideal just described, since the ideal is generated by homogeneous polynomials of degree $n$.

There is though a different way of making the set of nilpotent matrices into a scheme, namely, by realizing that a matrix $X$ is nilpotent if and only if all the eigenvalues of $X$ vanish, and hence if and only if the characteristic polynomial of $X$ is the polynomial $\lambda ^n$. The coefficients of the characteristic polynomial of the matrix $X$ (except the leading coefficient) are therefore polynomials in the entries of the matrix $X$ whose vanishing describes again the locus of nilpotent matrices. This time, though, the polynomials generating the ideal are homogeneous of degrees $1,2,\ldots,n$. It is not difficult to argue that in this case the ideal generated by these polynomials is indeed radical, so that the scheme thus defined is reduced.