It is an empirical question, as was the previous one.
As for data so far, right now we have two studies showing a smaller effect size for ACT than for a traditional CBT procedure done outside of an ACT model. Zettle, 2003 found a smaller effect for ACT than for systematic desensitization with trait anxiety when treating a relatively minor problem (math anxiety). The effect was the same in the area of math anxiety per se.
The second study is Forman, E. M., Hoffman, K. L., McGrath, K. B., Herbert, J. D., Bradsma, L. L., & Lowe, M. R. (2007). A comparison of acceptance- and control-based strategies for coping with food cravings: An analog study. Behaviour Research and Therapy, 45(10), 2372-2386. They compared ACT to a traditional CBT program for those who were impacted at different levels by food. 98 participants with chocolate cravings were exposed to a well known CBT-based protocol (Kelly Brownell’s LEARN program) and an ACT-based protocol or no instructions and required to carry chocolate with them of for two days. Those highly impacted by food related cues ate less and had fewer cravings in the ACT condition. But those not impacted by food, did worse in the ACT condition than in the CBT condition (and both did worse than doing nothing at all).
It may not make that much sense to use a procedure like ACT with minor problems because the issues it raises are so fundamental. This could be proven wrong with data. But note that in the Zettle study, ACT worked better with highly experientially avoidant subjects than with low avoidant subjects; desensitization did not show that relationship; in the Forman study those with high food impact were helped by ACT but not those with low food impact. It is not a comparison to CBT but another study [Masuda, A., Hayes, S. C., Fletcher, L. B., Seignourel, P. J., Bunting, K., Herbst, S. A., Twohig, M. P., & Lillis, J. (2007). The impact of Acceptance and Commitment Therapy versus education on stigma toward people with psychological disorders. Behaviour Research and Therapy, 45(11), 2764-2772 found that ACT was not better than education in reducing mental health stigma significantly among participants who were relatively flexible and non-avoidant to begin with. For experientially avoidant and inflexible participants, however, ACT was much better.
It seems possible based on this line of reasoning that CBT might actually work better than ACT in more confined and minor areas; in more severe or chronic areas ACT, might work a bit better ... and perhaps all of that because more avoidant and inflexible folks will be more dominant in severe areas. That is not a conclusion -- it is too early to say -- but it is a guess based on a few studies so far. And remember that ACT is part of CBT -- when we say "CBT" here we mean CBT methods placed into the context of a traditional CBT model that basically focuses on the idea that thoughts cause emotions and behavior.
It helps to keep in mind that ACT is a model not just a package. All of the behavioral methods and some of the cognitive ones can easily be put into ACT protocols. They are still ACT when that happens if they fit the model. That is especially true with behavioral methods -- which are a node on that hexagon model of ACT.
But ultimately we will have to show that, for example, exposure from an ACT perspective is better than (or at least works through a different process than) exposure from a traditional CBT perspective. We have a couple of small studies that indicate that might be true (e.g., see Jill Levitt’s dissertation in Behavior Therapy, 2004) but not large RCTs on the question. Some of these are underway right now (such as in Michelle Craske’s lab) so in a few years we will know.
Is there much in traditional CBT that is helpful? Yes, of course, and virtually all of what is known to work at the level of components fits with the ACT model so these procedures can be used from an ACT perspective. The things that contradict an ACT model are not known to work as components, such a cognitive restructuring. But even there you can modify it to be more a matter of cognitive flexiblity.
Is the ACT model a better place to put all of these procedures? Let’s see. The answer will probably not be “yes, always.” Presumably it is more likely to be “sometimes yes, sometimes no.” But both the yes and no answers will move us forward, and that is the whole point, not silly name brand struggles.