The greatness-Answering Amen is in one aspect even greater than saying the blessing:
Whoever answers Amen is [in one aspect] even greater than the person who said the blessing. Accordingly, we rule that whoever stole a Mitzvah and its blessing from another, a fine is not given if the “owner” was able to answer Amen. On the other hand, regarding another aspect, the person who says the blessing is greater than the person who says Amen. Thus, the reward of one who says the blessing and one who answers Amen is equal. Nevertheless, Hashem hastens the reward of one who says the blessing more than one who answers Amen. Accordingly, we rule that if a person has the choice between saying the blessing or answering Amen he should choose to say the blessing.
The reason: The reason Amen holds an advantage over the blessing is because the Amen is the finalization of the blessing, and is similar to the last warrior who won the battle. The Amen helps make the blessing become effective and draw down its Divine request. The Amen destroys any prosecuting angels who prevent the blessing from becoming true. It also banishes the forces of evil from nurturing from the blessing. A blessing is similar to unnotarized document, and the Amen is like a notary to the blessing that it will take place. The Amen is the second witness, who joins the first witness which is the person saying the blessing, to authenticate the blessing. Alternatively, the reason is because one who hears a blessing is as if he said it, and since in addition he answers Amen, he is therefore greater than the person who said the blessing who may not say Amen. Alternatively, the reason is because in the word Amen is hinted two names of Hashem, both Adnai and Havayah. Alternatively, the reason is because answering Amen includes three Mitzvos: Shavua, Kabala and Emuna.
 See Shach C.M. 382/7; Shaareiy Teshuvah 215/2; Likkutei Sichos 35/216-22
Background: There is an apparent dispute in the Talmud [Brachos 53b; Nazir 66b] as to what is greater; the Amen or the blessing. Each side of the argument seemingly carries a practical ramification in which they would dispute [regarding fine for stolen blessing, and regarding if one should try to lead Zimun]. The Shulchan Aruch in different areas ruled on these ramifications in a seemingly contradictory way. The Poskim [M”A 201/6; Shach C.M. 382/7] who answered this contradiction conclude that Amen is equal to the blessing, but not greater, and therefore the two rulings stand true. The Rebbe in Likkutei Sichos 35/216-22 takes an approach that in truth Amen is greater in one aspect and less great in another and therefore they equal out in reward.
Practical ramification-One who stole a blessing: The practical ramification in whether we say that one who answers Amen is greater than the person who said the blessing is regarding if one stole a blessing from another, such as someone else covered the blood of a bird, must the stealer pay the “owner” ten Zehuvim if the answered, or person could have answered, Amen. Practically, we rule that he is not obligated to pay being that answering Amen is even greater than the blessing. [Rama 382/1; Shach C.M. 382/4 [see there in great length]; Rosh in Chulin end of Perek Kisuiy Hadam in name of Rabbeinu Tam; Mordechai Chulin 655; Tur Y.D. 28; Smeh C.M. 382/7]
Practical ramification-Leading the Zimun: We rule that it is a Mitzvah to lead the Zimun and say the blessing over the Kos Shel Bracha. [Brachos 53b; Nazir 66b-end; Michaber 201/4; Admur 201/5] This implies that we do not rule like the statement in Brachos that answering Amen is greater than saying the blessing. [Simple implication of Gemara ibid that this matter is a dispute, and only according to the opinion who says it is not greater do we hold that one should lead the Zimun; Ravaya Brachos ibid; Piskei Harid Brachos ibid; Kesef Mishneh in opinion of Rambam Brachos 1/11] This contradicts the previous ruling regarding not giving a fine to one who stole a blessing. [M”A 201/6; Madanei Yom Tov on Rosh Chulin 6/8, brought in Shach C.M. 382/7] Some therefore answer that even the opposing opinion agrees that one who answers Amen gets rewarded, and that is why a fine is not given, and not because answering Amen is greater. [M”A ibid; Shach ibid; See Shaareiy Teshuvah 215/2; Panim Meiros 1/57; Birkeiy Yosef 215] Others answer that in truth everyone agrees that there is an aspect of advantage for the Amen over the blessing and an advantage of a quicker reward for the blessing over the Amen and therefore their reward is equal. [Madanei Yom Tov; Likkutei Sichos ibid p. 217]
 Madanei Yom Tov on Rosh Chulin 6/8; Likkutei Sichos 35/216-22 in answer of contradiction brought in coming footnotes
The reason: As he completes the blessing. This is the advantage of Poel over Koach. However, the person who said the blessing is greater in the fact that he started the blessing. [See reasons and coming footnotes and Likkutei Sichos ibid in great length on this subject]
 Brachos 53b; Nazir 66b; Shach C.M. 382/4 in initial explanation [see there in great length and his later explanation]; Tur Y.D. 28; Smeh C.M. 382/7; Levush 215, brought in Shaareiy Teshuvah 215/2; Rosh in Chulin end of Perek Kisuiy Hadam in name of Rabbeinu Tam regarding a stolen blessing; Mordechai Chulin 655; Implication of Semag Aseh 27; Orchos Chaim Dinei Amen; Rashal Hachovel 60; P”M 124 M”Z 4 “So we rule”; See Likkutei Sichos 35/216-22 in great length on this subject!
Other opinions: Some Poskim rule that although the Gemara makes the above statement, nevertheless it is not the final Halachic ruling, and rather we hold like the other opinion in the Gemara ibid that one who answers Amen is equal to one who says the blessing in reward, although Hashem hastens the reward for the person who said the blessing. [M”A 201/6; Shach C.M. 382/4 in later explanation; See Shaareiy Yosef 7 in name of Ritva, brought in Shaareiy Teshuvah 215/2; Panim Meiros 1/57; Birkeiy Yosef 215] See Likkutei Sichos ibid who answers the apparent contradictions and conflicting opinions
 Initial explanation in Shach C.M. 382/4 [see there in great length]; Rosh in Chulin end of Perek Kisuiy Hadam in name of Rabbeinu Tam; Mordechai Chulin 655; Tur Y.D. 28; Smeh C.M. 382/7; Likkutei Sichos ibid
Other opinions: Some Poskim rule that nevertheless a fine is imposed for the stolen Mitzvah. [Rashal Hachovel 60, brought in Shach ibid and negated by him] Some Poskim rule that this Halacha has nothing to do with the above statement of “Amen being greater than the blessing” as even the opposing opinion agrees that one who answers Amen gets rewarded equal, and that is why a fine is not given. [M”A 201/6; Later explanation in Shach C.M. 382/7; See Likkutei Sichos ibid who negates this explanation]
 Likkutei Sichos ibid in answer of contradiction brought in previous footnotes
The reason: As he started the blessing. This is the advantage of Koach over Poel. [ibid]
 M”A 201/6; Shach C.M. 382/7; Brachos 53b; Nazir 66b-end; Likkutei Sichos ibid
 Admur 201/5; Brachos 53b; Nazir 66b-end
 Admur 201/5; Michaber 201/4; Brachos 53b; Nazir 66b-end
 Admur 167/3; Rama 167/2; Darkei Moshe 167/4; Or Zarua based on Yerushalmi; Rosh end of Nazir; Gra 167
 Rebbe Nihoraiy in Brachos 53b and Nazir 66b as explained in Tosafus ibid
 Maharsha Nazir 66b; Likkutei Sichos ibid p. 220; Thus we rule that the person saying the blessing must concentrate on the Amen of the listener. [Admur 167/14; Rama 167/2; Or Zarua 1/102 and 178 based on Yerushalmi Brachos 7/3; P”M 167 A”A 36; Likkutei Sichos 35 p. 220] Furthermore, we find Poskim who rule that if the listener did not say Amen then neither the person who said the blessing or listener fulfill their obligation! [Or Zarua 1/102 and 178, brought in Darkei Moshe 167/4] We do not rule this way. [Darkei Moshe ibid; Likkutei Sichos ibid footnote 44]
 Or Hatorah Bereishis 3/566
 Orchos Chaim Din Aniyas Amen; Abudarham Seder Shacharis Shel Chol; See Zohar p. 229 and 271; Rokeiach 330;
 Rabbeinu Bechayeh Beshalach 14/31
 Perisha 124; See Admur 201/5; Likkutei Sichos ibid p. 218-219 who negates this explanation of Gemara
 Sefer Chassidim 19
 Shavuos 36a; Elya Raba 124/11; Perisha 124; This means that he is accepting the blessing with a swear and faith. See Chasam Sofer 15; See Likkutei Sichos ibid p. 219