Nearly every sport has specific shoes that allow the best performance. So does weightlifting.
Wearing weightlifting shoes during weightlifting or other strength sports offer two main advantages:
1 – The built-in heel lift makes it easier to perform a technically correct squat, snatch, cleand and jerk and bench press.
The raised heel allows for more forward knee travel. This means you can reach better squat depth, and/or have a more vertical torso while squatting. A more vertical torso lowers stress on the lower back. This allows you to use more weight, especially if your lower back is your weak point. For Olympische Weightlifting, the same mechanism allows you to catch the weigth in a lower position. For the bench press, the raised heel allows for a better back arch, while still maintaining full contact with the floor.
2 – Weightlifting shoes allow you to exert more force into the barbell, which makes for a more effective training session and better performance.
This happens because of two reasons:
2.1 The increased stability allows the muscle to exert more force.
Research has shown that muscle force output increases in a stable condition (source). Quality weightlifting shoes are very stable. The heel is made from materials that don’t budge under big weights (longitudinal stability). The metatarsal strap and heel cup provide lateral stability. Weightlifting shoes also usually have flat soles made out of rubber that provide very good grip.
2.2 Less force is lost
The heel and sole of a typical weightlifting shoe hardly compresses under stress. Therefore, more force is transmitted.
Running shoes for instance, are compressed a lot to absorb forces on purpose.
Enough reasons to get weightlifting shoes. However, most likely, you won’t immediately improve your performance much when wearing weightlifting shoes for the first time; you’ll first have to get used the slightly different execution of your exercises.