Jiu-jitsu? Judo?

Is it spelled Jujutsu? Jujitsu? Ju-Jitsu? Jiu-Jitsu?  
Yes. It has been translated by different people with different native languages over the years. There is some consensus that the japanese characters 柔術 translete to "Jujutsu" as the best spelling, but because earlier versions have been in use over time, they are likely to persist. How important is the spelling? Anything worth doing is worth doing well, even spelling, but the way you do your martial art is a lot more important (to me) than the way you spell it.  Wikipedia entry for Jujutsu

What is the difference between Jujutsu and Judo?  
Judo is a relatively modern martial art, founded by Jigoro Kano in the late 1800s. It is basically Jujutsu with the "dirty stuff" taken out and with a set of rules that allows for full speed competition with less chance of injury. Judo is an excellent way to see if your jujutsu works. 

Judo is also an excellent martial art, in and of itself, although it is not as complete a system as many forms of jujutsu, since strikes, weapons, smaller joint manipulation, etc. have been removed. What Judo has left when those things are removed is stuff that really works against an unwilling opponent. That makes Judo a worthwhile pursuit. Wikipedia entry for Judo