IS, ISIS and ISIL all refer to the same jihadist group, it’s just those outside that refer to them with different acronyms.

ISIS stands for Islamic State in Iraq and al-Sham, sometimes referred to in the West as Islamic State in Iraq and Syria.

ISIL stands for Islamic State in Iraq and the Levant (a bigger area of the Eastern Mediterranean including Lebanon, Syria, Jordan, Israel, Palestine)

According to the BBC it was the translation of al-Sham that raised inconsistencies. At first, news outlets were unsure how to translate it into English, as it was not immediately clear what the group was actually referring to. Al-Sham can be translated variously as "the Levant", "Greater Syria", "Syria" or even "Damascus".

The jihadist group’s goal is to restore an Islamic State, or caliphate in the region that includes Iraq, Syria, Jordan and Lebanon. Obama’s administration thinks ISIL is more accurate so Cameron followed suit.

The French call them Daesh

This link sums up the different acronyms:
http://www.independent.co.uk/news/world/middle-east/isis-vs-islamic-state-vs-isil-vs-daesh-what-do-t...