I work in a barn where we have both. I much prefer the concrete floors with thick rubber mats. The concrete stalls we do put deeper bedding in but overall they stay much cleaner and smell less. Therefor the stalls with concrete floors are kept for the seniors and very young horses. On a regular basis or after a horse has been sick we can disenfect the whole stall without making a huge muddy mess.
The dirt floors do absorb liquid and have to be maintained a lot more. They will settle & compact as to not be level and I do not feel they can ever be properly sanitized. Which tends to give me a case of the ickies if one gets perticularily wet and muddy for some reason.....i swear it is the only time I am ever "weird" about germs
P.S. I love the look of of the european barns