I know that the lance was mainly used by cavalry units, while the spear and pike were infantry weapons, but what were the key differences between them? Was it the length of the weapon, or the shape of the head?
The difference between a spear and a pike is that a spear is wielded with one hand (typically the right), leaving the other hand free to hold a shield. A pike is much longer, and is therefore a two-handed weapon. It cannot be balanced and used with just one hand. A lance is, as you say, a cavalry weapon, often longer than a spear but also generally wielded with one hand.
For each weapon, all other features could vary dramatically. In the Ancient world, spears could be anywhere between 5-8ft in length; some had a straight shaft while others are tapered (where the shaft becomes smaller in diameter towards the point for better balancing and reach); some had butt spikes while others had only a single point. The design of spearheads varied even within any given region in any given period of history, although variations on the leaf-shaped spearhead have been common since before the Iron Age. Here's a nice side-by-side of a Greek leaf-shaped spearhead (left) and butt spike (right).
As for pikes, in the Ancient world they ranged between 14-21ft in length; again, the design of their points, counterweights and shaft sleeves (possibly intended to hold together two halves of the long shaft, which were separated for transportation) varied a lot. However, the heads of pikes tended to be smaller than those of spears, and their butt spikes heavier and more elaborate, to preserve balance. Since the weapon was gripped closer to the back end, weight had to be concentrated there to allow it to be wielded easily. This replica shows the difference between the butt spike (left) and the leaf-shaped head (right) of a Macedonian pike. The ribs on the butt spike serve no purpose except to add weight.
Most of the time whether "lance", "spear", or "pike" is used typically comes down to the whims of the translator. The actual terms used tended to change from period to period and author to author, and usually historical writers tended to be less interested in specific terminology than many modern historians are.
For Elizabethan military treatises typically: the long, 10-20 ft infantry weapon is the "pike", the shorter, usually ~8 ft weapon is the "half-pike", the "lance" is a heavy cavalry weapon sometimes almost as long as a pike, the "spear" is used by a type of light cavalry, sometimes called "spearmen", light spears meant for throwing are called "darts", and the "javelin" is a type of long thrusting polearm similar in length to a pike.
