You will get a bit more effective range and downrange energy out of the .308.
i would use it on smaller-medium sized game. I wouldn't trust it on elk, but I believe I could take a small-medium deer. Not sure what type of deer you have over there. I have seen some hunting shows about Roe Deer, those would be a very acceptable sized deer for .223.
For deer that are over 200lbs adult weight, I would want the .308. From the 150-200lb deer, shot placement is going to be important, but a good shot will do the job. Some species of deer get so large that I would want something bigger than .308 to hunt.
What species might you be going after if you decide to go hunting?