Rising is usually something we talk about a physical object. Like an action. Like the waves are rising. The temperature is rising. Your leg is rising. Increase has more to do with numbers. Increase the number of dogs and add them with the total number of cats. You usually use it in math or even in work. Like the calls are increased by the evening.

As nouns the difference between rise and increase:  RISE- is the process of or an action or instance of moving upwards or becoming greater . INCREASE -Is an amount by which a quantity is increased.

“Rise" specifically indicates elevation. "Increase" indicates the addition of.