If the Scripture is taken literally then the Sabbath was in existence before the Mosaic Law:
27 On the seventh day some of the people went out to gather, but they found none. 28 And the LORD said to Moses, “How long will you refuse to keep my commandments and my laws? 29 See! The LORD has given you the Sabbath; therefore on the sixth day he gives you bread for two days. Remain each of you in his place; let no one go out of his place on the seventh day.” 30 So the people rested on the seventh day. (Exodus 16:27-30 ESV)
Therefore the Mosaic Law extended what was already in existence:
10 but the seventh day is a Sabbath to the LORD your God. On it you shall not do any work, you, or your son, or your daughter, your male servant, or your female servant, or your livestock, or the sojourner who is within your gates. (Exodus 20:10 ESV)
What was initially an ordnance of individual responsibility was extended to include others. In other words, prior to the law a person could observe the Sabbath by their own actions; after the Law a person might also be responsible to see that others, including animals and sojourners within their gates observed the Sabbath.
To rest means not to work; to follow the Sabbath means to ensure the others you are responsible for join you in rest. In may be overly simplifying but the primary distinction is corporate rest in contrast to individual rest.
Now if someone is a child of God, then there will be a corporate rest with God, who is responsible for you. So for the people of God, there remains a Sabbath rest (with God) that He will ensure.