What's the point of being able to put the power token on the character? If you have no power tokens but lose a power challenge you still lose that power token which is placed on the character with "infamy", right?

Im just trying to see what the differance in having it on your house card as opposed to having it on a character with "infamy" is.

Only power on the house card can usually be taken, that is the specific place that power challenges look to take power from. Infamy keeps your power safe from such challenges and counts to your total (because the rules for infamy say to do so) at the cost of making that power more vulnerable in general as if the character dies the power is lost as well.

Like renown it keeps power away from the house card ,although unlike renown it doesn't add more power, it just gives an option as to where it can go.

There can be benefits - especially with the new brotherhood characters (When not running the brotherhood agenda)

Its a bit of a balance between preventing opponents claiming power from your house vs the vulnerablity of the character

To reiterate: winning a POWER challenge let's you take power counters from a player's House card. If there is no power on the House card, you get nothing for winning the challenge - even if they have power counters on characters. Power on Infamy cards are safe from opponents winning POWER challenges against you.