The obvious advantage of traveling alone is that you and only you determine the daily schedule. There are no compromises to be made and no adjustments for anyone else. I can remember traveling in Paris and Italy and covering a tremendous amount of ground in just a few days that would have taken me much longer if I had been traveling with a partner. I also enjoy the sense of anonymity and freedom.
In terms of safety, I think risks increase a bit when traveling alone in Europe. Let's face it, there are some bad people out there and I get the feeling they will target someone traveling without a partner before an accompanied traveler. Therefore I would tend to always stay with a crowd, be ultra careful at night doing things like eating at the hotel restaurant or nearby instead of scooting across town. I am hesitant to enter a public restroom unless there is an attendant or other people in it. Unfortunately, women traveling alone must be even more careful.
The other issue that bothers me when traveling without a partner is dining. To solve that, I would go to the well-known Jim Haynes dinner in Paris to meet other travelers. See:
http://www.jim-haynes.com/
I have also posted on sites like TripAdvisor and craigslist to look for dining companions when traveling alone. However, I urge caution when doing so - only meet in a public place with a group of people.
Churches and other similar organization often sponsor meetings and group dinners. I have always felt welcome there. In fact, this would be my top tip for traveling solo.
I think the reason my travelers like my small group tours is that the itinerary is flexible and we always plan a group dinner every night. Lots of people say a touring situation like that is just what they were looking for.