You can't have a certified diamond graded as G-H. It's either G or H. Certified means that it comes with a laboratory report.
If you're referring to the Iliana range, they are not certified diamonds. They are graded in-house. So some of the Iliana range will be designated G and some H so they market them all as G-H in colour. Grading for colour is done with the diamonds face down and using a reference set however if it's done in-house there is always the concern that they won't be as strict in grading as a laboratory. Having said that, that's a general comment and NOT aimed at TJC. With the exception of 1 or 2 pieces, all of my Iliana diamonds have been very good indeed and have, in some circumstances, been assessed as higher than G-H and some have been VVS in quality.
In terms of what you see, the difference is minimal unless you are super sensitive to colour. G and H colour diamonds are normally white looking with very little trace of yellow and the price difference is minimal. Price depends on size, clarity, cut and colour so an H may well cost more if it has a better cut (light return) etc. G-H diamonds are generally considered to be more than acceptable.