I noticed that while typing out my previous post that Firefox was not automatically spell checking as I was typing. Coming from Chrome which does this automatically I thought this was something that should work out of the box.
It looks like you need to have a dictionary installed, language pack installed and the option to check spelling ticked.
Make sure you have spell checking enabled. Menu > Options > search for spell check.
Once you know you have that turned on then you can check to see if you have the dictionary installed. Visit the Dictionary and language pack add on page. Scroll to your desired language and and install the dictionary. Give Firefox a quick restart and spell checking will now start working.