The distinction might be that the object is the motive of the act, either in the sense of the ultimate goal of the act or in the sense of what the agent has in mind while performing the act. On the other hand, the determining ground could be what determines whether or not we perform the act. If we consider a person who rescues a drowning baby, Kant might be saying that the object is the highest good in the sense that the object is what the person is considering while they perform the rescue, or in the sense that the person's ultimate goal of acting selflessly is the highest good. Dutifulness, on the other hand, is what determines whether or not the baby will be saved, and also whether or not these other mental attitudes will be held.
It is also possible, though, that Kant would say rather that the highest good is the object, in the sense of being the conscious aim, while dutifulness is the determining ground, in the sense of being the ultimate goal. It is also possible that none of these distinctions is quite what Kant has in mind.
In order to clarify the notion of an antinomy, let us consider one of the antinomies from the first Critique. Events in the world are always caused by other events in the world. The antinomy of freedom asks whether there is a first cause. If there is, this is a problem, for it is itself uncaused, so there must not be one. If there is not one, then we must comprehend an infinite series of causes as having already taken place, and since we cannot do that, there must be a first cause after all. The solution is to distinguish the noumenal and the phenomenal. The first cause exists, but only in the noumenal realm and there is no problem with noumenal uncaused causes. And yet it is not true that there is an infinite sequence of causes, for the phenomenal extends only so far as one happens to have had experiences of it. Since we grasp a finite but indeterminately extendable amount, there is no question of an infinite uncaused series.
Now we can anticipate what is to come. Good actions depend on the highest good to make them worthwhile. Assuming that there is a highest good leads to paradox, as does assuming that there is no highest good. The solution lies in reference to the noumenal world.