Hi there!
There are 2 main differences between rise and increase
The first one is a slight difference in meaning:
Increase means to become greater in value or size.
Rise means to become higher or move upwards.
Therefore, as I have just seen that Igor has already mentioned, the sun can rise but not increase.
There are many situations where you can use both 'rise' and 'increase' These are often situations where you can imagine the size of something on a graph or chart, for example:
The population can rise and increase - it can move upwards (on a chart), and get bigger in value
The temperature can rise and increase - it can move upwards (on a chart), and get bigger in value
Our profits can rise and increase - it can move upwards (on a chart), and get bigger in value
The other difference is that you can increase something, but you cannot rise something:
'I will increase the price of the product' - correct
'The price of the product will increase' - correct
'I will rise the the price of the product' - incorrect
'The price of the product will rise' - correct
With rise, something can rise only on its own, someone else cannot rise it. Here is another example:
We are increasing the number of homes available - correct
The number of homes available is increasing - correct
We are rising the number of homes available - incorrect
The number of homes available is rising - correct
If you want to 'rise' something, we have to use a different verb, the verb 'to raise'. Here are some examples from earlier:
'I will increase the price of the product' - correct
'The price of the product will increase' - correct
'I will RAISE the the price of the product' - correct
'The price of the product will RISE' - correct
Hope this helps, let me know if you have any questions!