The fact that all $\beta$-functions vanish in a theory is equivalent to the statement that the energy-momentum tensor is traceless by the operator identity
$$
T^\mu_\mu(x) = \sum_\mathcal{O} \beta_\mathcal{O} \mathcal{O}(x) = 0.
$$
This is in turn sufficient to conclude that the theory is conformal: the charge
$$
K^\mu \equiv \int d\Sigma_\nu \left( 2 x^\mu x_\rho T^{\nu\rho} - x^2 T^{\mu\nu} \right)
$$
associated with special conformal transformations is conserved.
In $\mathcal{N} = 4$ super Yang-Mills, all $\beta$-functions vanish, and therefore the charge $K^\mu$ is conserved and the theory is conformal.
In a theory that is scale invariant but not conformal, not all $\beta$-function vanish: there is a particular operator (or a linear combination of operators) that is the divergence of a current $V^\mu$ (called the
"virial" current) and whose $\beta$-function is non-zero:
$$
T^\mu_\mu(x) = \partial_\mu V^\mu(x) \neq 0.
$$
You can check that the charge $D$ associated with scale transformations,
$$
D \equiv \int d\Sigma_\nu x_\mu T^{\mu\nu}
$$
is conserved in this case, which means that the theory is scale invariant. However, the charge $K^\mu$ is not conserved: the theory is not conformal.
The other possibility for a theory to be scale invariant but not conformal is to have no energy-momentum tensor. In this case, the construction of the charge $K^\mu$ fails from the start. But we know that $\mathcal{N} = 4$ super Yang-Mills does have an energy-momentum tensor, since we can construct it explicitly from the Lagrangian.
