Both are of course connected, but denote very different concepts. Your definition of variance is fine with me. It tells us, on average, how much the flow deviates from its average.
Intermittency is much harder to explain. I'll do my best here:
In a scale invariant flow, the $n$'th order correlation function will take a scaling form,
$$\Delta\Psi(n,r) = \langle \left[\Phi(x+r,t)-\Phi(x,t)\right]^n\rangle = \langle \left[\phi'(x+r,t)-\phi'(x,t)\right]^n\rangle \sim r^{\zeta_n} \, .$$
If the statistics of $\Phi$ are gaussian, all the correlation functions are determined by the second one, $\Delta\Psi(2,r)$, and we have
$$\zeta_n = \frac{\zeta_2}{2}n \, .$$
All the scaling exponents are linearly related. Note that Kolmogorov theory uses this assumption and the fact that $\zeta_3 = 1$ to get $\zeta_n = n/3$.
In general though, the statistics of $\Phi$ are not Gaussian and this does not hold. Intermittency is the phyisical process that leads deviations of $\zeta_n$ from $n\zeta_2/2$. It is called "intermittency" because it can be traced down to the tale of the probability distribution of gradients of $\Psi$, i.e. to large but rare fluctuations which do not appear in a Gaussian distribution. These happen intermittently because they are so rare.
