I am thinking of running Fibre optic from one room to a Nearby ONT/Router, I currently have about 15-20 meters of Cat 6, With a Gigabit UFB internet plan. With a short distance of around 20 meters would there be a huge speed gap between Cat 6 and Fibre?
The equipment interfaces determine the speed, albeit the medium used may have restrictions on which interfaces it can be used. If you have 1 Gbps ethernet, it doesn't matter if it is copper or fiber, it will be 1 Gbps ethernet.
Properly installed and tested Category-6 cabling can even run 10 Gbps over a short distance. You cable vendor should have provided your company with the lists of the category tests and the results for each cable installed. If the cable passed all the Category-6 tests, you should have no problems running 1000BASE-T (1 Gbps ethernet on UTP).