2 types: Casein & Lactose
There are 2 totally different types of intolerance:
An issue with lactose, the natural sugar in milk. This is very rare in Irish children and a little more common but still quite rare in Irish adults. (Approximate incidence in Irish population has been estimated at 1% but this may be higher now due to greater numbers avoiding all dairy long term which actually causes lactose intolerance)
An issue with the actual milk itself, its protein makeup, known as casein.
If you suffer from lactose intolerance (the rarer issue), you will suffer mostly with liquid milk but could be fine with a little aged cheese (where the lactose has mostly been fermeted away). With casein intolerance, you will likely not do too well with either milk or cheese but cheese would likely be worse.