How co-related is it to BTC?
GBTC, the Grayscale Bitcoin Trust, is simply a company that owns bitcoins. If you own shares of GBTC, you own a portion of the trust and therefore you own that portion of the trust's bitcoins.
The advantage of owning GBTC rather than owning bitcoins directly is that you can hold the shares in a brokerage account and trade them in the stock market. In the U.S., the ability to hold bitcoins in a retirement account is very restricted, but the ability to hold GBTC shares is not.
The disadvantage is that you are relying on GBTC to hold the bitcoins for you, potentially increasing the risk of loss or theft. Additionally, when you buy shares on the open market you are paying a premium for the bitcoins that they represent. Furthermore, Grayscale Investments (different from GBTC) is paid a fee of 2% per year from the trust to manage the trust.
If the advantages don't apply to you, then there is no reason to own shares of GBTC instead of bitcoins.