Exceptional leave is granted to people who don't qualify for asylum but to qualify for leave to remain under the Human Rights Act.
TA is granted when someone is being refused entry but is temporarily allowed to physically enter the UK for a number of days (usually)...this is usually pending their return home or onward trip to another country, or, if the IO wants to consider their application to enter further. Someone with TA may be physically in the UK but legally they are not!
Vicky
