The LR44 is an alkaline battery, while the 357 is a silver oxide battery. The 357 battery lasts longer than the LR44 and has a more stable voltage level over time. The 357 battery costs more than the LR44 as well.
The 357 and LR44 batteries both turn chemicals into electricity, but they do it in different ways. The silver oxide 357 battery works better than the alkaline LR44 battery. The voltage of the 357 battery stays around 1.55 volts until the chemical energy in the battery is almost gone, at which point the voltage drops quickly. The voltage of an LR44 alkaline cell keeps going down as the battery dies.
Even though the milliamp-hour ratings given by the manufacturer may look more similar, the 357 has about twice the useful capacity of the LR44. Even if more electrochemical capacity remains, the LR44 can lose voltage too quickly for a device to work properly.
Most devices can use either the 357 or the LR44 battery, but devices that need a constant voltage, like precision callipers and some watches, benefit from the 357’s constant voltage performance. Devices that need more power can use the 357’s larger useful capacity to their advantage.