'Arise' means 'begin', 'appear', 'come to one's notice'. It is used mostly with abstract nouns as subjects.
A discussion across about the best way to pay.
I'm afraid a difficulty has arisen.
'Rise' usually means 'get higher', 'come/go up'.
My hopes are rising.
What time does the sun rise?
Prices keep rising.
Note that we usually say that people get up in the morning. 'Rise' is only used with this meaning in a very formal style.
'Arise' and 'Rise' are irregular verbs:
(a)rise (a)rose (a)risen
