"Why do we worship on Sunday, when the Bible says the sabbath is Saturday? And does it make a difference?"
The first commandment explicating the sabbath day is found in Exodus, "Remember the sabbath day, and keep it holy. Six days you shall labor and do all your work. But the seventh day is a sabbath to the Lord your God; you shall not do any work you, your son or your daughter, your male or female slave, your livestock, or the alien resident in your towns. For in six days the Lord made heaven and earth, the sea, and all that is in them, but rested the seventh day; therefore the Lord blessed the sabbath day and consecrated it" (20.8-11). From this verse the Israelites developed some 613 rules/laws prohibiting work on the sabbath (the seventh day, Saturday, which began on Friday at sundown and ended Saturday at sundown). The term "work" was specifically defined in these 613 laws and all of Israel were expected to refrain from any sort of work or from causing anyone else to work.
However, in the early understanding of the Israelites the sabbath was not necessarily a day set aside specifically for worship. Although according to Numbers 28.9-10 there were prescribed sabbath sacrifices made on behalf of the nation, it is apparent the typical worshiper was expected to bring personal offerings on days other than the sabbath (cf., Leviticus 23.15-16). Indeed, herding or carrying an animal to the sanctuary would have been considered work. In any event, there wasn't an emphasis on weekly worship during the years of the first temple (approximately 1000 to 400 BCE), for Israelite men were required to go to the temple for worship only four times each year (Leviticus 23).
By the first century CE it is clear weekly worship had become an accepted custom. For one thing, synagogues had become rather common and most Jewish families lived within a sabbath's journey (the maximum walking distance allowed on a sabbath) of their local house of worship. Indeed, it was Jesus' custom to attend the local synagogue on each sabbath wherever he visited (Luke 4.16).
However, after Jesus was crucified there was a movement by some of the local Christians to gather on the first day of the week, Sunday, to worship and share communion (Acts 20.7). Although it isn't directly stated in scripture, it seems clear the notion of worshiping on the first day of the week was in remembrance/honor of the resurrection day of Christ. However, the Jewish Christians also celebrated the sabbath in the prescribed way. Indeed, it is inconceivable that they would not have been, since there was such vehemence by the church in Jerusalem to require adherence to the whole Jewish law upon all who embraced Christianity.
That is, until Paul took Christianity out of the sphere of Jewish influence and into the Greco-Roman world. There, when a large number of non-Jewish Christians began meeting on "the Lord's Day" instead of keeping the sabbath laws, there was an outcry from "the church." In his letter to the Romans, Paul addressed this problem. In essence, he told the Roman Christians that they were not bound to worship on any particular day: "Some judge one day to be better than another, while others judge all days to be alike. Let all be fully convinced in their own minds" (Romans 14.5).
Modern Christianity has generally chosen to put aside worship on the sabbath day in deference to the Lord's Day. However, these same Christians have historically applied the ancient sabbath laws to Sunday (no work, etc.). The original purpose of the sabbath was to provide a day of rest. Indeed, Jesus said, "The sabbath was made for humankind, and not humankind for the sabbath" (Mark 2.27). To insist that any given day is more holy or important than another is to impinge upon the beliefs and traditions of others a clear violation to the rule of tolerance as defined by Paul (Romans 14).
So according to scripture, the day of worship, whether Saturday or Sunday is unimportant. However, it is important that each person have a time of rest and have the opportunity for worship.