If we use your own definitions,
By actual possibility I mean the possibility which is implied by ability or power.
By logical possibility I mean whether concepts of reality contradict each other or not.
then I would say that yes, it is straightforward to have something which is logically impossible but actually possible. The reason for this is that you do not specify which concepts — or models — of reality you are concerned with.
For a half-serious example, consider the dramatic revelation in Star Wars: Episode V (The Empire Strikes Back). Luke's reaction to the revelation that Darth Vader is his father is disbelief: that it is impossible. Whatever emotions drove this reaction, it is likely that this was his reaction to the fact that the assertion contradicted his mental model of Darth Vader and Anakin Skywalker being seperate individuals, one of whom literally killed the other, as Ben Kenobi described in Episode IV. Relative to this model of reality, it was logically impossible for Darth Vader to be Anakin Skywalker, because it violated an assumption used in constructing the model. But if we accept what is asserted or demonstrated by the later Star Wars films, it was in fact actually possible, and indeed true.
More seriously, discovering things which are actually possible but which are logically (more accurately, theoretically) impossible, is a good approximation to how science works according to Popper: by falsification. If a theory predicts that something should not happen (or is impossible to make happen), but which subsequently does happen, this invalidates the theory. For this reason, I would prefer to call this "theoretical", rather than "logical", impossibility, because it places the fault clearly where the failure of the model can more easily be remedied — by improving or replacing the theory.
In modern science, where we accept that probability may be an unavoidable feature of physical theories, we are presented with a more complicated situation. When a theory predicts one probability distribution, but experiment produces another, has the theory been falsified even if we assume that the experiment was "executed perfectly"? With probabilities, there is always of course a non-zero chance of freak occurrances in which events drawn from one distribution resemble another. This is of course less likely, the more random samples you take, but in most cases you cannot actually rule out the possibility that one distribution will in practise, with a finite number of samples, produce the curve of another. We then move from impossibility to improbability — where we might ask whether something is actually probable while being theoretically (or logically) improbable. This is a somewhat more nuanced, but still essentially Popperian notion of falsifiability: we accept that there can be such events, and that when they arise they indicate a failure of the theory.