Many experts favor closed-chain over open-chain exercises. They do so under the assumption that they are safer and more functional.
In reality, our bodies can use both options naturally.
You should choose exercises, whether closed-chain or open-chain, that best allow you to load all the major muscles and joints safely. This depends more on properties of the exercise unrelated to this distinction.
Chains
Sometimes experts will refer to each category as closed kinetic chain vs. open kinetic chain. A kinetic chain just refers to a series of body parts. The hip, knee, and ankle would form a lower extremity kinetic chain. Our bodies function as units and not as bits and pieces. Action that occurs at one link in the chain will affect the rest of the unit. The addition or absence of the word kinetic makes no difference to the issue.
With an open-chain exercise, the body remains in place and the limb performing the action moves. You overcome the resistance.
With a closed-chain exercise, the body moves and the limb stays fixed. You fail to overcome the resistance, so your whole body moves instead.
The dip and push-up are closed-chain exercises. The bench press is an open-chain exercise. The squat is a closed-chain exercise. The leg press is an open-chain exercise. The pull-up is a closed-chain exercise. The row with a dumbbell is an open-chain exercise.
Conclusions
As a rule for lifting, open-chain works best for the upper body and closed-chain works best for the lower body. This allows you to heavily load the right movement patterns. It occurs this way though because the best exercises in each category just so happen to go in one category or the other, not because closed or open matters much.
Some believe closed always means better. They support bodyweight gymnastics exercises like ring pull-ups and dips. Both these exercises can harm the shoulder. Gymnastics in general has quite a reputation for how harshly it affects this joint. This represents a peril of adhering to the closed-chain only path for its own sake. Consider that running and pushing or pulling objects in everyday life would function as open-chain exercises.
Weighted push-ups, dips, and pull-ups, which are upper body closed-chain exercises, are not automatically poor choices and can work well in some cases. Each of these exercises has unrelated problems. Weighted push-ups limit the weight that can you can add in the long run. Pull-ups can hurt the shoulders while working the biceps less effectively since you pull vertically. The dip will also hurt the shoulders for most trainees.
Still, in many cases, the muscles used will look exactly the same for the largest muscles for each category. Some argue that closed-chain exercises activate the core more so but it depends. A push-up will activate the core more than a bench press. The squat activates the core more so than a leg press. An inverted row would not necessarily activate the core more so than a row with a dumbbell.
Within the context of a complete program, the lesser core activation on the bench press versus the push-up matters little. Sometimes the core can become the limiting factor, such as on an inverted row, to limit the stimulation to the much larger muscle groups. It depends on the exercise examined.
Many isolation exercises are open-chain and rely on machines. Most closed-chain exercises are compound exercises and rely on free weights. It ranks much more importantly that you choose compound exercises over isolation. You also should use free weights over machines.
Many will incorrectly lump all open-chain exercises as isolation. The bench press is an open-chain yet compound exercise. It also is the best possible pushing movement we could choose.
If each category does have slightly different muscle activation patterns then it still matters little. If you wanted to transfer strength gained from your squat to biking, you must ride. Skills are either specific or not; strength training should remain general.
Avoid the Closed-Chain vs. Open-Chain Exercise Distinction
Some will say to include both. They feel comfortable taking the moderate position which may seem more comprehensive. This would just invite redundancy. It also allows the possibility of poor exercises entering into the fray. Variety of its own sake serves no purpose.
I suggest the barbell bench press, the barbell squat, and the dumbbell row as the three safest and best ways to strength train all the major muscle groups. If you add intervals with resisted sprinting, stairclimbing, or jumping, you take care of some of the stabilizers less stimulated through the lifts while also including specific cardio. These four form a well-rounded and efficient program.
They are a combination of closed-chain and open-chain exercises. No exercises were picked with this distinction in mind. Ignore the closed-chain vs. open-chain exercise debate.