This is an extension of Timothy Chow's answer about $SO_n$ behaving differently depending on the parity of $n$. Even when $n=2m$ (i.e., type $D_m$), the groups behave differently depending on the parity of $m$.

Since the center of $SO_{2m}$ is order $2$, we know the center of $Spin_{2m}$ is order $4$. But is it $\mu_2^2$ or $\mu_4$? The answer is: if $m$ is even, its center is $\mu_2^2$, while if $m$ is odd, its center is $\mu_4$.

Another example: $O_{2m}$ has nilpotent orbits (i.e., orbits of $O_{2m}$ acting on nilpotent elements in the Lie algebra $\mathfrak{so}_{2m}$) parameterized by partitions of $2m$ with an even number of even parts. In $SO_{2m}$ most nilpotents remain a single orbit, but those $O_{2m}$-orbits corresponding to partitions with only even parts becomes two $SO_{2m}$-orbits. Such phenomenon only occurs when $m$ is even.