The function of the two structures is the same.
When you want to MAKE A SUGGESTION about what you (and the people with you) should do, you can choose any of these expressions.
But the FORM is different:
Shall I + VERB?
- Shall I open the window?
- Shall I turn off the TV?
(Meaning: Do you want me to open the window/turn off the TV?)
Shall we + VERB?
- Shall we visit Aunt Lily?
- Shall we buy it today?
(Meaning: Do you want us to visit Aunt Lily? Do you want us to buy it today?)
What about…/ How about… + NOUN (or verb+ing)
- What about asking Joe?
- How about Friday?
(Meaning: Do you think it’s a good idea to ask Joe? Is Friday good for you?)
– What shall we do today?
– How about visiting Aunt Lily?
– It’s hot in here.
– Shall I open the window?
– When shall we meet?
– How about Friday?