A servant works at someone’s home, often doing lowly tasks. You might feel like a servant when your mom insists that you do the dishes before you can do anything fun, but think how she feels cooking dinner every day!
If you’re a servant, you serve someone else. In that 1930s mystery novel, chances are the rich family had servants to set the table and do the laundry. Nowadays, calling someone who does those things a servant has an insulting connotation. If you think your hard work gets no respect, you might feel like a servant. But people who work for the government like to be called "public servants," because they serve the good of public and not themselves.