SSN is your identity in US. Need SSN for getting a phone, car, house, etc as well as bank accounts, credit cards and so it is pretty important. Not relevant in context of new H1B except that the visa officer might expect it given you had a past job visa.
In general, what is the purpose of the SSN and is there any use for the SSN?
With the context of H1B1, same question as above: Is there any use for the SSN in the context of a new H1B1 application?