I think you are not giving complete information for your question. Usually, we use different tension for 2 main reason. First is different string structure e.g. poly or multifilament etc. Second is personal requirement e.g. increase hand feel etc. If you are using hybid string, poly will require a lower tension (5-10% less than normal). And some people will put poly string as main (Most of the people). But still some people will put it as cross (Federer). So for this reason , people will put different tension on the same racquet. But for spin, power & control, it is not a main factor. But Overall lower tension will give you a better chance to general more spin & power, but less control.
I agree with Michael. But there's something I'd like to add to that. As Michael's mentioned, in a hybrid setup, you should string the poly with the lower tension (usually 4-6 lbs lower than your favourite one-piece multi setup). But whatever you do, don't put a multi (e.g. Babolat XCel, or Tecnifibre XOne Biphase, or what have you) on the mains and a poly on the crosses. It just creates toy-like bouncy feeling. Put the soft string on the mains only when it's natural gut ( haven't tried natural gut in my life so far but I hear people say that).
As for answer your main question of the difference in spin/power/control, you should always remember in your hybrid setup, the mains string is the dominant string. That means you get most of the feeling of your racket from the mains. The crosses are only there to improve, or tweak this feeling. For instance, putting a poly on the mains means a lot of spin, control (and durability). However, a one-piece poly setup feels quite hard and less powerful (even if you string low). So to soften this feeling, you put a soft multi (or gut if you can afford it) to make the overall feeling softer. On the other hand, if you put a soft string on the mains, you get comfort and power. But you lack control and spin, and durability. So you put a poly on the crosses to rectify this.
Natural gut is supposed to be the ultimate tennis string, which has all the attributes: spin, control, power, comfort... anything but durability and value. That's why Roger Federer puts a high-end natural gut on the mains. But he thinks he needs even more control, so he puts a Luxilon poly on the crosses to push the control even further.
Of course, how we string our rackets is totally up to each of us. Using Rafa's setup (which I heard is different from what is generally believed on the web, especially TW) doesn't make you a new world number 1. Even pros have different string and tension selections. I'm afraid you have to try it yourself to tell the difference, my friend.
Actually, the ball doesn't know how to distinguish which is the Mains and Crosses. And the ball is always impact with the stringbed with Mains and Crosses at the same time. Then the main point is that you can only feel the difference between the stiffness of the stringbed, and it is the mix of both mains and crosses.
SO, what are the difference is that the nature of a tennis string-job, is always go with the Mains before doing the crosses. Then the difference come from here, as the mains are strung at free, which the tension should be same as the tensionhead reference tension (neglect the friction from grommets, and angle of pulling). However, after the mains are done, when doing the crosses, then weaving is now getting into picture, and these weaving will give a lot of friction when pulling the crosses. In general, this friction can lower the actual strung tension on string down to approx 70% or even lower.
So, as you mentioned, with a 60/56 and 56/60 (actual on string tension 60/39 & 56/42), the difference should be small enough to be neglectable in term of power.
However, when you look at it in spin potential then the differnece should be more noticeable. As most of the time, when you applying spin, brushing i should be along the horizontal plain, and the Mains should sliding along the Crosses in order to give longer ball bite dwell. Then if the Crosses actual on string tension is lower, then the Mains are relative looser lock into position, then should give you a better spin and bite. That's why, when the strings, both Mains and Crosses, are notching, and losing tension with give a very uncontrolable feeling. The ball will fly out of the stringbed and not enough bite on the ball for spin.
The kind of tension loss cause from weaving only happens when doing a "hard-weave", but not so much on "soft-weave". But it's very simple to rectify this problem. Simply pull the hard-weaving cross string twice. To be precise: pull the string, don't clamp, release the tensioner, pull the string again, then clamp. There, you have your accurately pulled cross string. I tend to pull twice even on some supposedly-soft-weaves. That way I can narrow down all the unnecessary worries, as, in theory, you can never overstretch a string if the tension head's tension is set right., no matter how many times you pull.
Yes, you are right, double pulling is one of the effective ways narrowing down the friction from weaving. But it's undoubtedly weaving is affecting the actual on string tension.
This is relativity issue, as if both 60/56 & 56/60 applying the same stringing method (single pull or double pull), the outcome will somewhat similar. However, double pulling will certainly giving you a stiffer stringbed.
That's why, stick to a one machine and stringer will give the consistancy. Luckily, I string for my own since from high-school.