You will get priority check-in and lounge access.
Baggage allowance on LH is more weird, and do not depend on fare class, but on fare option (Light/Classic/Flex):
On K light fare no checked bags are allowed.
On K classic 1 checked bag is allowed + 1 for *G.
Carry-on is allowed.
You will not be able to book T class on LH.com if K-class is available. It is possible to book it by phone, but there is ridiculous additional fee, so the best way is to use a travel agent.
There is no difference between T and K class regarding *G benefits.