I have played the game.
Happiness gives you the pact, and if you already have a pact with the demon, you get some cards instead.
Eagerness just gives you cards.
To make a pact, you need to max out happiness. However, I believe you want the EXACT method of making a demon happy. It's different for each demon.
Some demons have different traits. To find these traits out, click "Contact" and hover over the demon you want to contact. Its traits will be shown. (These being things like "Foolish" or other such things.)
Pick a character (or characters) with traits that seem to attribute to the demon. (If the demon is foolish, try to act like it's smart and such)
Mostly, since it's quite difficult to get the right combination, I pick one character and try all their options, it should increase happiness with one of them. If it doesn't, I try a different character or a combination of characters.
Tatsuya (or the MC, I just kept his default name) is usually the best one for contacting demons. His contact options usually raise one or two of the demon's stats. Just test each one until you find the option that raises happiness for that demon, then spam it till happiness is full.
Personally, I find it's best to try and raise both happiness and eagerness when you have a pact, as you'll get pretty good cards.
If you make the demon scared or angry, it will break your pact and run away or fight.
Hope I helped
P.S. As for the advantages to a pact... I don't know myself >.< I haven't played very long, only 1 or 2 hours.