Rule of thumb: if it's a word composed of several kanjis, it's more formal. As for the difference between 直す and 正す, the former is "correct" as in "fix", and the latter is "set right". Use ALC to figure out the nuances: http://eow.alc.co.jp/search?q=%e6%ad%a3%e3%81%99&ref=sa

Hi, They are the translations of "corregir" of Spanish. The difference in my opinion is that 添削する and 訂正する are a bit more formal and 直す is used the most. You can use 直す and 正す for other cases not only "to correct what you write/say", for example, 服装を直す/正す、悪いくせを直す/正す. I would translate "please correct my English" 私の英語を直してください though you can use all of four words in this case. I hope this was helpful.