In America at least, the word "slave" has a connotative meaning that conjures up images of slavery prior to the Civil War, where people were beaten, mistreated, sold indiscriminately, and many other horrors.
In the Bible, slavery was much different. For one thing, a slave would only be a slave for six years and then had to be set free. When he was set free, the master had to provide him with a means to live on his own. There was also a provision that at the end of the sixth year, the slave could say, "I love my master and do not wish to go free." If he did so, he could remain with his master. This type of slavery must have been extremely different than slavery in America to end up with slaves volunteering to remain slaves.
So, the word slave may be appropriate if we didn't have the images of people being beaten, mistreated, killed, and all sorts of things like that. The word servant seems to lend itself more accurately to the actual situation of slavery in the Old Testament, and therefore the likely intent of Paul and others who wrote the New Testament and referred to be a slave/servant of Christ.
With Jesus as our Masters, who, out of His love, laid down His own life for us, our response to Him would most certainly be that of the Jewish slave who says, "I love my master and do not want to leave" rather than a slave from an American plantation who would have run away at any opportunity that arose.
So, servant seems to convey the Jewish idea much better than the word slave because of our cultural context.