You can naively compute ALL-SAT in $2^npoly(n,m)$ time, where $n$ is the number of variables and $m$ is the number of clauses. You clearly need $\Omega(2^n)$ time in the worst case just to write down the assignments (in the case of a tautology). If the strong exponential time hypothesis (SETH) holds, then this is not much harder than SAT itself, so under SETH the complexity of SAT, ALL-SAT, #SAT is the same (up to polynomial factors).
Moreover, without SETH you can claim that given access to a $\#SAT$ oracle, you can output all satisfying assignments in time $k(\varphi)poly(n,m)$ where $k(\varphi)$ is the number of satisfying assignments for $\varphi$ (just see if the number of satisfying assignments changes when substituting $1$ or $0$ to some variable $x_i$). This also shows that in some sense ALL-SAT is not much harder than $\#SAT$.