The difference is mainly the amount of certainty in each sentence. "The bill should be paid before the end of the month." suggests that the bill isn't paid yet but the speaker thinks it will be before the end of the month. Or it could be a suggestion. Maybe this person has a deadline for that bill for March 2th but someone is suggesting that they pay the bill before the end of the month to avoid paying it late. Either way there is a measure of uncertainty. It is unknown if the bill will be paid.
"The bill has to be paid before the end of the month." is more definite. Whatever that bill is, it MUST be paid before the month ends. It is more of a command than a suggestion like the first one.
Hope this helps!