In the case of QCD there is asymptotic freedom, meaning that though the theory is strongly coupled at low energies (such that we still cannot analytically calculate how the atomic nucleus stays together) the coupling becomes less and less as we go to higher and higher energies. This means that the ultimate picture of the behavior of quarks is a weakly coupled theory where the quarks hardly interact with one another anymore.
In the case of the Higgs self coupling (or any other theory which becomes non-perturbative / strongly coupled / non-unitary) at higher energies it means that at higher energies the theory isn't suitable for any calculations (as opposed to QCD which still is in this case). This means that at higher energies certain calculations will give $\infty$ as an answer and no longer make any sense, we will find quantum mechanical probabilities that add op to more than $1$. Thus people hope (and historically this has happened a few times) that these theories that are non-unitary will become unitary again by introducing additional fields that will regulate the high energy behavior.
In the case of QCD this isn't a problem as we can go to higher energies where the theory does make sense. In the case of QED the self coupling of the electron was a problem, which was only solved by the additional interactions introduced in the Standard Model (though, that has some problems of its own as you pointed out)