Here's my take on the differences between the two:
People who are on or are in need of 'fat loss' are those who have little to no exercise background, have higher BF levels with little lean body mass, and need to strip down size before even attempting or worrying about increasing muscle.
There are many phases to go through when doing a fat loss diet. But overall, it mainly comes down to an individual needing to learn WHAT to eat, and also portion control. Most people that need a fat loss diet are people that have little nutritional knowledge, and generally have over eaten for a period of time. So it's more about 'smarter' choices with their food. Which after time, they can move into a more advanced calorie and macro tracking program to further their progress.
Since individuals in need of fat loss programs tend to be the more sedentary type, cardio is a must. Generally it entails just getting out and doing SOMETHING on a regular basis. Burn some calories.
Lifting for these programs is essentially an extra spur in the calorie burning process. Maintaining what muscle is already there is a plus, but it's not always a main point for the program. These people aren't always familiar with lifting, so they tend to start off with what the more advanced lifter would consider fluff work. Reps tend to fall in the mid to high range. It tends to be a familiarization with the gym, the movements, the form, all that fun stuff.
'Cutting' is for the individual that already has a decent muscle base, is more experienced in lifting, and has a body fat that is already lower by 'average' standards. These people are wanting to 'lean out' to show more definition in the muscle they've already built.
The cutting diet is much more precise. Calories and macros are tracked tightly. And the macro amounts are based more on preserving the muscle rather than just being in a caloric deficit.
(And I'm going to disagree with TALO on the 'no fat' approach to cutting. I've done both ways, with fats and without, and I feel that keeping some fats in there helps in a lot of ways. But, it IS based on the individual)
Being that people who are cutting can't make drastic cuts in the caloric intake (due to the fact that they can only go so low with them, and there's the issue of preserving muscle still), cardio is the main way to add in a little extra calorie burning when needed. But this should still be kept to a minimum. Losing fat through 'cutting' should primarily be done through diet (again, with the muscle sparing... too much cardio compromises the muscle mass).
Overall training volume is reduced slightly due to the caloric restriction (to avoid over training and compromising recovery time), but emphasis still remains on doing compound movements, and working in both strength and hypertrophy ranges (with the occasional fluff rep ranges depending on the program). Continuing to lift regularly while cutting is important because... it's muscle sparing! (noticing a trend here?