You can think of "cutting" or "bulking" workouts as additional supplements to your actual cutting or gaining phase. Just like how you would use extra carbs for bulking, you use "cutting" workouts for losing weight. While there aren't any black and white rules for separating the two, there are some important factors in these workouts that will greatly enhance your progress.
Now, when you are cutting (from a diet perspective), you must understand that you will NOT gain muscle. Unless you are on some heavy hormones that is. This is a fact, if you are in caloric deficit, you can not put on muscle mass. With that in mind, let's break down the goal of "cutting" quickly.
First, you want to maintain as much mass as possible. How is this achieved? Well, in traditional bro science, people will tell you to do high reps and high sets with lighter weights to "tone" the muscle. This is one of the biggest fitness myths in the history of the industry. It's been shown in hundreds of studies this is not true in any way, you can not "tone" muscle, you can either make it bigger or smaller (I'm sure you already know this but I'm trying to address a general audience). So then, how do we maintain the most muscle? Well, when you use light weights and high reps, you're basically telling the muscles you don't need them because you're not recruiting all the muscle fibres and motor units that you could be. So, the answer is rather obvious now, the most important thing for maintaining muscle mass, is maintaining STRENGTH (you can ever get stronger on a cut!!). When you maintain lifting heavy during a caloric deficit, you're telling your body "I really need these muscles in order to keep lifting these weights", as a result your body is less likely to break down those muscles for energy during a caloric excess. So, key point: lift heavy weights on a cut.
Second, you want to aid in fat loss by burning extra calories, working your metabolism more. Traditionally, the easiest and most common way that this is done is through reduced rest times between exercises. For example, on a bulking workout you want to maximize tension to prove muscle growth, so you can sacrifice rest times for lifting heavier weights, i.e 3-4 sets of 315lbs bench press for 8 reps, with 2-3 min rest times is perfect for bulking. Now, on a cut however, we know that we're not going to gain muscle and we just want to give our metabolism a boost. So if we reduce our rest times between exercises and sets, then your heart rate will be elevated more often and for longer durations, and as a result you will be tapping into your metabolic energy system and burn more calories. For example, on a cut, (After the main strength move I talked about above), you want to limit rest times to max 60-90 seconds, and gradually reduce them. It's important to note that gradually reducing your rest times is also a means of making progressional overload and will only make you a better lifter in the long run. Key point: reduce rest times gradually.
In terms of volume, this is highly variable and usually depends on the individual. Dorian Yates, a 6x Mr Olympia was known to do 10-15 sets per workout while Jay Cutler, another Mr Olympia was known to do 10-15 sets per EXERCISE. Generally speaking though, the more volume you have the more calories you will burn, but consequently it'll be harder to recover, so it's a double edged sword. In terms of progression, in a cut, you want to progress on reducing rest times and maintaining (and even improving) your strenght, since its impossible to gain mass. In terms of exercises, you obviously want big strength movements that will burn big calories and also make you stronger, the definition of compound movements. Doing calf raises on a cut isn't going to do much other than make you better at calf raises...
Now on a bulk, we want to maximize muscle tension and overload, and keep progressing. In order to keep progressing, it is a good idea to periodize your workouts and introduce various shock techniques such as dropsets and supersets so your body never adapts and you can keep growing. Now, unlike the cut, the rest times can be as long as you need in order for you to lift the most weight for the most reps (max tension). With volume, like I said, it depends, do enough volume to feel that your muscles are reaching failure but make sure it's not so much that your muscles can't recover and don't grow efficiently.