Classic CBT (cognitive behavioral therapy) aims to modify maladaptive cognitions by supplying more adaptive alternatives, while ACT (acceptance and commitment therapy) assumes the mind to be a meaningless chatterbox that should be habituated, not focused on, or manipulated.
For example, when a patient is afraid of flying, the classic CBT approach would be to teach him more realistic risk evaluation, which is minimal in this case, a manageable way to interpret his physical over-arousal, as expectation of danger – not actual danger, and behavioral exposure as means to learn that there is no real danger.
The ACT approach would not argue with the patient's risk assessment in flying and in his panic, claiming that these serve avoidance, and that the patient should just listen passively to the arguments in his mind like they were an open channel radio with other people talking, until the volume fades, and he can just go through the motions, just do what he should do, disregarding his anxious thoughts and feelings.
If we put aside the contradictory theoretical frameworks, we can find that a pragmatic similarity would be the behavioral "just do it," and the difference is whether to tackle specific maladaptive cognitions or circumvent them. A combined practice could be to direct the patient to exposure, tackling the grossly maladaptive cognitions that lead to avoidance, and circumventing debates where cognitive debating is a cause for avoidance. Alternatively we could use CBT when we can offer a more adaptive cognition and ACT when the cognition is sensible but nevertheless distressful. Another distinction can be to tackle the ego-ayntonic cognitions, which the patient tries to justify with arguments, while circumventing the ego-dystonic cognitions, which admits have no sense behind them.
In my view, psychotherapy's curative effect is replacing dogmatic responses with a soft dissonance that allows variability of response – not providing a more adaptive response but facilitating adaptivity itself. Classic CBT should not provide a right answer - "you know flying is 10X less dangerous than driving" - but provide the patient a grain of salt to his preconceptions - "did you ever question your conception that flying is dangerous?" - show him that another way is possible. In that way, ACT should not ask the patient to give up the meaning of his thoughts ("what were the numbers? You see? Meaningless, and you're stuck with them!") as much as it should alleviate intensity of the patient's internal debate regardless of the arguments - "where does all this thinking lead you?" – show that it can be harmful by itself. If the mutual goal of mental agility can be established, the therapist can vary his practices, being attuned to the patient's field of preoccupations, and estimating the patient's capacity to loosen up a variety of dogmatic mental structures. The combined practice can be focusing on specific cognitions as a start, and then generalizing the same principle to the entire internal discourse, giving the patient room for air. Unifying these practices under one goal could also allow us to measure their effects comparatively, and use them more appropriately.