What is the actual difference between complete and incomplete combustion, and why does it matter?

One variable decides it: how much oxygen is available. Everything else, including the products, the flame colour and the danger, follows from that.

The two equations

With plenty of oxygen, a hydrocarbon burns completely. Every carbon atom ends up as carbon dioxide and every hydrogen as water:

CH4  +  2O2   ->   CO2  +  2H2O

With a limited oxygen supply, there is not enough to take every carbon all the way to CO2. Some stops at carbon monoxide, and some does not oxidise at all and comes off as soot:

2CH4  +  3O2   ->   2CO  +  4H2O
 CH4  +   O2   ->    C   +  2H2O

Notice the hydrogen still becomes water in every case. It is only the carbon whose fate changes, because hydrogen oxidises more readily. If you remember that one asymmetry, you can reconstruct any of these equations rather than memorising three.

What the exam actually asks

Rarely "define the two". Usually something applied, where the marks are for connecting the oxygen supply to a consequence.

Flame colour. Complete combustion gives a clean blue flame. Incomplete gives a yellow or orange, smoky flame, because the yellow light comes from glowing soot particles. A question describing a yellow flame on a Bunsen is telling you the air hole is closed.

Energy released. Complete combustion releases more energy per mole of fuel. The carbon in CO or soot still has chemical energy that has not been released, so an incomplete burn wastes fuel. That is why a yellow Bunsen flame heats things more slowly.

Danger. Carbon monoxide is the one that matters. It is colourless, odourless, and binds to haemoglobin far more strongly than oxygen does, so it stops blood carrying oxygen. That is why faulty gas appliances in poorly ventilated rooms kill people, and it is the single most commonly examined application of this topic.

The mark most people lose

Questions ask you to explain, and candidates describe. "Incomplete combustion produces carbon monoxide" is a description and scores little. The explanation is the chain:

Limited oxygen → not enough to oxidise all the carbon fully → some carbon oxidises only as far as CO → CO binds to haemoglobin in place of oxygen → the blood cannot carry oxygen to tissues.

Each arrow is a step the mark scheme is looking for. Write the chain out rather than jumping from the cause to the final consequence, because the marks sit on the links, not on the endpoints.