I recently had a bit of a revelation while paying particular attention to how my balance worked depending on whether I took the next step with my foot toe-first or heel-first. It gave me all sorts of ideas for things to give people to think about we more often than not simply take for granted.
But before I cover the difference between each step let's think about the back foot for one moment.
Whether it is for stepping in and out of measure or whether it is for lunging, how we distribute our weight from the back foot can be a matter of losing or winning a bout. Generally speaking in both cases you want to delay the shift of weight from the back foot to the front foot as much as possible. This allows you space and energy to divert your step at the very last second thus permitting a void or a retreat to happen depending on what your opponent does. This control of feet is often rather overlooked, especially in more "upright" fencing systems. However, in many rapier systems, which can have some rather dramatic weight-shifts, this is a very important principle. One needs only to read between the lines of Girard Thibault d'Anvers's treatise to realise that if you allow your weight to be delayed over the front foot you can then respond in proper tempo to an incoming attack and step accordingly for a safe defence to occur. Many Italian systems also allow for this space by having the major guards with the weight placed further back onto the back leg.
With this in mind, whether one steps or whether on lunges, which part of the front foot will make contact with the ground first is essential. When stepping with the toes extended, it allows for greater balance and flexibility due to a wider circle being created between the back foot, which acts as a pivot, and the front foot, which "draws the circle". This is the sort of step you want to make when simply stepping in and out of measure, whether you are circling your opponent or moving along a straight line. As an exercise, stand feet together, the back foot (left if you're right handed) either firmly on the ground or with the heel raised a little off the ground and take a step forward with toe pointed towards the ground. Now feel the weight staying over your back leg and see how stable you are without putting the front foot down on the ground, instead drawing an arc with your toe.
Now let's see what happens with the lunge. An important aspect of the lunge is the ability to recover. Some later systems advocate deep lunges and they have their reasons to do so. However focussing solely on rapier systems, it is fair to assume that one wants to be able to safely recover from a lunge. As we've seen the front toe pointing down increases the diameter of your "stepping circle". With the lunge, the further forward you step, the harder it is to recover. I personally recommend lunges to be no longer than one foot's length, that's the distance of your own foot, not the imperial measure. And to allow for this shortened distance, I recommend your lunging step to be lead with your heel. Not only will it allow you to step at a safe recoverable distance but it will also prevent more severe ankle twisting which usually happens due to the ball of the front foot rotating on landing. As a reminder, the foot must always be the last part of the body to start moving during a lunge, lunges should always be lead by the point of the sword to minimise the distance it requires to travel to its intended target.
To summarise, when stepping, lead with the toes, when lunging, lead with the heel. And while this is a simple deduction borne of my own experiments, I am more than happy to have people discuss the practical application of this method, bearing in mind that I make no claim of universal truth along all fencing systems, more that it is mostly relevant amongst rapier systems, but may be helpful for other systems.