Drunk Prayer vs Drunken Grace after Meals

 The Gemara in Berachos 31b states that one is not allowed to pray whilst he is in a drunken state

אל תתן את אמתך לפני בת בליעל אמר רבי אלעזר מכאן לשכור שמתפלל כאלו עובד ע''ז כתיב הכא לפני בת בליעל וכתיב התם {דברים יג-יד} יצאו אנשים בני בליעל מקרבך מה להלן ע''ז אף כאן ע''ז

The Gemara says that if one does pray in a drunken state it is as if he is עובד ע''ז.

It sounds severe to say that it is such a bad Aveira that it is like עובד ע''ז. Understood, it is not respectful or even downright rude to appear before the king in a self induced compromised state. However why would the Gemara compare it to Avoda Zara? 

Furthermore Tosafos wuotes a Yerushalmi (Terumos 1:4) that states as follows:

ירושלמי במסכת תרומות פרק קמא רבי זירא בעי קמיה דרבי אמי שכור מהו שיברך א''ל ואכלת ושבעת וברכת ואפילו מדומדם. פירוש ואפי' הוא שתוי ושכור כל כך שאינו יכול לדבר כראוי דהכי משמע ושבעת וברכת

That one is not only allowed but obligated to Bentch even if he is drunk. Why would there be such a significant difference between praying and bentching to the extent that praying drunk is like avoda zara but there is still a chiyuv de'oraisa to bentch even while drunk?

The root of Avoda Zara is אני ואפסי עוד, meaning that it is all about "myself". I am willing to serve God as long as I get to decide how. Also, it has to serve MY purpose. The person serving Avoda Zara is really serving himself and is doing it all to satisfy his own needs. I will pray to any God that will get me what I want. That is Avoda Zara....אני ואפסי עוד. 

Understanding it in this context can help explain why davening while drunk would be considered Avoda Zara. The person davening does not care about respecting God as long as he gets to ask for his needs. That is the ultimate in selfishness because he is using God to get what HE wants. That is the root of all Avoda Zara.

Bentching on the other hand is all about thankfulness. We are not asking for anything by bentching we are showing our gratitude. Someone who is drunk who thanks somebody is actually showing more gratitude than one who is sober because it comes from within without any ulterior motive. 

The difference between the Halachos of Tefilah and Birkas Hamazon is because they are different in their essence. Tefilah is in its essence bakasha and Beracha is in its essence gratitude. A drunken request is disrespectful and selfish while a drunken "Thank You" is respectful and unselfish.