What is the benefit(s) of Praying with a Minyan? Besides for all the additional things you can now do with a Minyan (Kaddish, Kriah etc). By prayer, I mean the Shmoneh Esreh. What is different in saying Shemoneh Esreh alone or together with nine other men?
The Rambam writes (Tefillah 8:1)
תפילת הציבור, נשמעת תמיד; ואפילו היו בהן חטאים, אין הקדוש ברוך הוא מואס תפילתן של רבים. לפיכך צריך אדם לשתף עצמו עם הציבור, ולא יתפלל יחידי, כל זמן שיכול להתפלל בציבור...
The prayer of the community is always heard, even if there were sinners among them, for God refuses to reject the prayer of many. Therefore one should always combine himself with the community and not pray alone whenever he is able to pray with the community.
Additionally, the Gemara (Brachot 8a at the top) brings a number of drashot regarding prayer with the congregation, calling it an Eit Ratzon -- a time of Divine favor, and some other positive adjectives.
Some sources expressing the advantage of davening with a Minyan (I am making this a community wiki so anyone can add more sources):
Gemorah Berachos 8a: "Rabbi Yochanan said in the name of the Rashb"i: What is the meaning of the verse 'My prayer is to You Hashem at a favourable time'? When is it a favorable time? At the time the congregation prays . . R' Nassan said: From where do we know that the Holy One, Blessed is he does not despise the prayer of the congregation? It is written: 'Behold G-d does not despise the numerous.' It is written: "He redeemed my soul in peace from battles' - the Holy One Blessed is He said: 'Whoever engages in Torah and acts of kindness and prays with the congregation, I consider it as if he has redeemed me and my children from amongst the nations of the world."
Sanhedrin 39a: The Shechina rests on every assembly of ten Jews.
when there's a minyan the shechina (divine presence) comes.
when alone, angels must bring the prayer up, and its more problematic, and less acceptable. (zohar)
also, every amen you say in a minyan/kedusha has big effects in the mystical worlds, and brings bracha to this world, and can only be said when the shechina comes to a minyan
"Who is like G-d who answers us whenever we call out to Him?" (Devarim 4:7). But isn't it written "Seek G-d [only] when He can be found" (Yeshayah 55:6)? This one is dealing when with a congregation; this one is dealing when alone.