They are different in this following aspect.
Leak Off Test – you pressure test shoe and formation until formation break down.
Let’s I explain more about it: Leak off Test is conducted in order to find the fracture pressure (fracture gradient) of formation and shoe. When conducting the LOT, you will pump drilling fluid to until you see the fracture trend of formation. Once formation is fractured, the first pressure that deviated from a trend is typically called Leak Off Pressure. We use the leak off pressure to calculate LOT.
Formation Integrity Test – you test strength of shoe and formation to designed pressure.
Let’s I explain more about it: Formation Integrity Test is typically used for testing strength of formation and shoe by increasing Bottom Hole Pressure (BHP) to designed pressure. When you do the FIT test, you will increase surface pressure until it reaches the required pressure only. There is no intention to break the formation with FIT. You will do FIT to ensure that you will be able to drill to section target depth and will be able to control the well in case of well control situation without underground blow out.