This is definitely going to be cell type and protein specific. Some proteins will degrade even at -80 (or from the freeze-thaw) and must be worked with fresh for W. blot analysis. For RNA extraction I suppose the same could be true, but I have never had an issue as long as I add some sort of protection reagent such as RNA later or RNA protect.
I agree with previuos answer, in my hand I always risospend pellet in Trizol if I was sure that I want the sample for RNA extraction. If I have doubt I pellet the cells and put at -80°C but only for short period. If you are sure that you will perform W. blot and RNA extraction prepare two pellet and risospend in Trizol and in you lysis buffer, it is so easy with cells pellet.
First You need to mention what type of cells that was homogenized and centrifuged.
For animal tissue cells usually people will add protease inhibitors along with Homogenization buffer. If sensitive enzyme is kept for longer time storage, it may denature. So better design the expt based on what type of protein u r looking at (protein in pellet or supernatant) and select specific antibody and continue with protein work
For RNA extraction I think it does not effect much.
--My area of Research --protein / enzymes studies from brain tissue and cell culture