I love off-peak season travel. Although I've never been to Europe in December, I have been there numerous times in November. Some smaller guest houses and hostels MAY be closed, but I can't speak from experience on Spain at that time. You may encounter a "winter" public transport schedule, but overall, with fewer tourists, I think you'll enjoy it.
A good guidebook will tell you what to expect as far as weather...and let me tell you, it's best to pay attention to that. I've been surprised more than once by how cold it was in Eastern Europe (for example) in the Autumn.
As far as Spain, the Mediterranean in general has a Wet/Dry climate, so it MAY be rainier than usual. Once again, check the particular region in Spain you're planning on going to...I'm sure it is different in some areas...
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"Life is not measured in the number of breaths we take, but by the places that take our breath away..."