Saito Sensei used to use the hanmi with the front foot turned more (toes pointing slightly out) - much like what you see in Yoshinkan Aikido. If you find a copy of the book ""Aikido - It's Heart and Appearance" by him you will see diagrams with foot positions that are quite intricate. In the 1980s when I went there the first time he saw some of us from Scandinavia doing the "old" hanmi with the front foot turned slightly outward (we had learned this from Tomita Sensei (based in Stockholm, Sweden), who had trained with Saito Sensei in the late 60s). He explained that there was nothing wrong with this but he recommended just keeping the front foot straight (when standing in hanmi). His reasoning was that with the front foot turned outward it was easy to move in the direction where the foot was pointing ( for instance in left hanmi - to the front left with the same foot) but moving forward in the other direction the foot was pointing away from that direction, making this movement slightly unstable.
But he always said that there was nothing wrong with the "old" hanmi. As Jacob pointed out - some techniques use a strong hip turn and here it feels natural to turn the front foot more outward. The basic idea is the hip turns the foot.
I like this idea of a straighter front foot enabling weight/energy/focus...maybe ki even... to be channelled in a straight line...with an uke in the line of fire I'm thinking this could be a way of aligning ones connection to them.
I've seen Phillipe Voarino discuss use of a greater turn of the foot...and suggesting that this engages the hips to a greater degree so as a positive many increase power of hip turn and enhancing cutting motion in sword work.
As a negative though, I feel that it puts extra stress and strain on ankle and knee joints...so 20 or 30 years of it could cause complications....and for me personally it detracts from the feeling of whole body alignment that I'm currently looking at.
So...not a comprehensive list...but maybe some reasoning for some differences between styles.