Answer:
The frequent mistake many of our fellow American travelers make is thinking of Europe as a big Disney World with individual countries as just so many Tomorrowlands, Frontier Lands, and Adventure Lands. They're not.Think of the countries you list as distinct, unique cultures, each of them worthy of much more than a cursory glimpse from the rolling deck of a cruise liner or from the window of the tour bus that is waiting at each port of call.
You will want to linger, to stroll through the back streets, to sit in the cafés frequented by the locals, not those pre-selected by the tour guide because they're run by his brother-in-law. What if a new place — or new friends in it — catches your fancy and you want to linger and explore more than a port call allows? On a cruise, the ship will leave without you. On your own Eurailling trip, you can just catch a later train.
Independent travel requires a bit of advance planning, which is almost as much fun as taking the trip itself. But the rewards of seeing exactly what YOU want to see (not the tour organizer) on your own terms and at your own pace, are so much more vastly rewarding than any packaged tour.
As one of the other answerers indicated, ten days simply are not enough. Remember, you will lose two or three of those days just getting to Europe and back. if you don't live in a major gateway city such as New York, Dallas, Boston, etc., you will have to fly to a gateway to catch the overnight flight to Europe. Then you will spend the next day recuperating from jetlag, during which you will be too exhausted to do much. Then you will lose another whole day returning to the U.S. Give yourself at least three weeks minimum. A month is better. Five or six weeks is awesome!
Cheaper? Yeah, the cruise lines discount their cabins all the time. A cruise may end up being a huge bargain. But will you really "see" Europe? No. You'll see just enough to whet your appetite and wish you had planned your own independent vacation. Save a bit more money and accumulate a bit more vacation time from work to see Europe properly, at your own pace.

{ 4 comments… read them below or add one }
With a cruise you will know beforehand how much you pay, as all but the excursions are included.
And the price of the excursions can be found for most cruises.
But mostly the cruises are not cheap, they are luxury.
Traveling by train you can keep your own budget, staying in hostels and eating in the cheapest kind of places or even do your own cooking from the cheapest supermarket food.
Or you can spend a lot more if you want to stay in fancy hotels and spend money on top restaurants.
If the places you want to visit are all accessible by cruise, in ten days it is likely the better option, as traveling overland will take a lot of your time, with a cruise you cover the miles in the night, with a train much of the traveling will have to be done in daytime. (There are some night trains, but mostly not when and where you need them.)
So while it may not be the cheapest option, cruise might be the best for you.
In general, I agree with Willeke on this. The cruise is likely to be somewhat more expensive, but also much more convenient especially if you want to see a new place every day. With the cruise, you won't have to be hauling your luggage on and off trains frequently, for example. You can generally do the shore excursions on your own cheaper than the prices offered by the cruise line, but there is some effort required for the planning and you need to take care to make sure you get back to the ship on time.
However, there are some useful night trains for some of the trip depending on exactly what you want to see in France, Spain and Italy. If, for instance, you want to visit Barcelona, Paris, and Venice, Florence or Rome, there are viable options for taking the trains during the night and you can get tickets fairly cheaply on routes to/from Italy if you buy them far enough in advance to get the discounted fares. There are night trains in each direction on the route between Paris and Barcelona, Barcelona and Milan (doesn't run every day), Paris and Venice, Paris and Rome (stops in Florence en route). If you want to visit other places, that would use more daylight time in transit.
http://www.renfe.es/horarios/english/index.html (Spain train site)
http://www.ferroviedellostato.it/homepage_en.html (Italy train site)
http://www.sncf.com/en_EN/flash/ (France train site)
The German train site is good for finding schedules throughout Europe: http://www.bahn.de/i/view/USA/en/index.shtml
The cruise is the best and cheapest way because 1: a part of Italy is seperated from the rest of the country, and 2: a cruise is better for the enviroment
Ten days are not enough to visit these 3 countries, no matter if it's by train or ship, but anyway I suggest the Eur rail pass by train.
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