OK the general difference is that NuBits are a currency, which should be used by everyone who wants to make use of a crypto currency without any price fluctuations. NuBits users can totally stop thinking about the system at this point and just should know that they can buy NBT for at most 1 USD and sell them for at least 1 USD.
NuShare holders form a central bank which is able to take influence on the amount of NuBits in existence. This can be done by the methods described in the white paper, i.e. grants to create NBT, parking to temporarily remove NBT, burning to permanently remove NBT. The creation of NuBits can hereby serve different needs, either to develop a product that helps Nu (like the grant you posted) or liquidity providers who want to get compensated. All these actions are performed in the block minting process, and you can roughly say that if you have X% of all NuShares that are staking, then you will provide X% of the voting mass.
This is the one and only task of the NuShare holders: Using the tools provided by the protocol to ensure the NuBits peg! NuShares itself are like a stock, and its success depends on how well this goal was established and as described in the whitepaper specific grants pay out dividends to shareholders.
