Lets say I have 10 different treatments and these 10 treatments can be considered combinations of 2 factors - as in a factorial design experiment. What do I miss if I treat this data as coming from a complete randomized design instead of a factorial design? What will be the consequences?
The randomization in a completely randomized design refers to the fact that the experimental units are randomly assigned to treatments. Even though a factorial design is very structured, you can still assign the experimental units to the levels randomly. This prevents bias due to the differences in your experimental units from being confounded with effects you want to observe. So, in answer to your question, there would be no consequences.
If you can't completely randomize the allocation of units to treatments but you can randomize it to some degree then there are sometimes ways to handle it in the analysis (see split-plot designs for example). Otherwise you're in a bad spot, and reasonable people might hold that any inference from your experiment is invalid.