How many fish were used to develop the relative condition factor... or did you use a published standard weight equation. Generally, you should never use Fulton's condition factor if a standard weight equation is available, because Fulton's is generic and usually length dependent, while standard equations are species specific and length independent.

Relative condition factor

Statistically you can compare but biologically it could be problematic. Because both factors has different assumptions. For instance if the species does exhibit isometric growth then you can compare both factor otherwise comparing both factors could be problematic.

I don't think you should do any comparison  between different kind of condition factors of the same species/population. It would be of no meaning biologically. Unless for example, your main purpose is on methodology , which  may tell you  which  type of condition factor is more suitable to use over others, for a specific  species. Any type of C.F. were theoretically assumed to be more efficient than others by us at the beginning, even if you could do some statistical comparison of them beforehand.

I would follow the general thinking of all my fellows colleagues. As assumptions of those two condition factors are different, it would not be sound to compare them. Even your first comparison don't seems useful for me. Just choose one of them and stck with it for all your other comparisons (among and within population, between groups...).  

I am not aware of relative condition factor and wouldn't use Fultons for the reason already provided. Same with your statistical test; it is not useful to compare them statistically. I'm not sure what critters you are working with, but I would recommend calculating relative weights (Wr) that you may be able to compare to standards or among populations if that is what you'd really like to do. You should also consider comparing size structure among populations (i.e., proportional stock density or PSD) or comparing growth among populations or age classes (i.e., relative growth index or RGI) if you have age and growth data. The attached article directs you to the PSD and Wr literature and shows how to calculate RGI.

Article Growth standards for nine North American fish species

Be careful! you may have a problem of collinearity when comparing the two. I think you should choose to which more information provided to achieve your goals.

I agree with most of the previous comments. The assumption for each of the indexes are not exactly the same, so, further comparisons may be problematic. Regarding methodological issues there are several approaches to meassure conditions. I recomend, among others, the following reads:  

Article On the misuse of residuals in ecology: Regression of residua...

Article On the misuse of residuals in ecology: testing regression re...