Comparison of acidities
Once you understand that the acetate ion is more stable the the ethoxide ion, we understand the fact that after releasing The acidic hydrogen the final product formed is more stable for acetic acid as compared to ethanol.
This would lead to the conclusion that acetic acid is a better acid as compared to ethanol.  

While comparing the acidity of two species, only the comparative study of conjugate base is enough, since its simply the comparison between compounds.  

Comparing the stability with the original compounds will lead you to the conclusion that whether the compound itself has a tendency to act as an acid or base.
Alternately one could compare the stability of the original compound and its conjugate(Which is the acid dissociation constant) and compare its value with the acid dissociation constant of other compunds to compare its acidity.  

In any case comparing the stability of the conjugate ions will tell you which one of them is comparatively more acidic/basic than others. It will not let you conclude whether its the nature of the compound to naturally act as an acid/base or not. For that one should compare the stability with original compund or just check its dissociation constant.