Griffins are mostly land predators
The lion tail is there for counter balance and helps with manoeuvrability on land.
A bird's tail helps with flight. The lack of one suggests that a griffin doesn't spend much time flying.
So, all this means that griffins with a lion tail spend more time on ground, especially hunting. The wings must be something they use seldom. They can still use it to go up or down a cliff for example.
One interesting idea is to use it for scouting - go up a tree or other elevated part of the terrain and then swoop down in the direction where the pray is. Close by but not exactly next to it. The griffin can then use the stealth and hunting tactics of a lion - approach and attack.
One more notable thing about lion's tails - the little puff at the end aids with communication. Lions can swing their tail to communicate with other members of their pride and griffins can also probably do that. It will be quite useful if there are some posted as lookouts up high. They can swing their tail to notify "Prey spotted" or "Everything clear".
This seems like a logical way to use their bodies and features.