At the beginning of Shemoneh Esrei we beseech Hashem:
(ב) אֲדֹנָי שְׂפָתַי תִּפְתָּח וּפִי יַגִּיד תְּהִלָּתֶֽךָ:
(2) My Master, open my lips, and my mouth will speak Your praise.
But at the end of davening we switch verbiage and ask Hashem:
(א) יִהְיוּ לְרָצוֹן אִמְרֵי פִי וְהֶגְיוֹן לִבִּי לְפָנֶֽיךָ יְהֹוָה צוּרִי וְגוֹאֲלִי:
(1) May the words (lit. sayings) of my mouth and the thoughts of my heart be acceptable before You, Adonoy, my Rock and my Redeemer.
It begs the question ,what did we just perform in our tefillos, a "speaking" or a "saying?" And what is the difference between the two terms.
Perhaps we can say that yagid refers to a more polished and concise mode of speech whereas amira refers to more of a discursive way of talking. (See Gra in Ishei Yisrael to Hashem Sifasai)
When we begin Shemoneh Esrei we are trembling before the King, and speaking with awe. As the tefilla goes on and we beseech Hashem for our needs, we begin to speak more freely, asking for specific things and pouring out our sorrows and hardships.
Therefore we start with "yagid" a term for more polished speech and end with "amar" a more discursive type of speech.
When we begin Shemoneh Esrei we are trembling before the King, and speaking with awe. As the tefilla goes on and we beseech Hashem for our needs, we begin to speak more freely, asking for specific things and pouring out our sorrows and hardships.
Therefore we start with "yagid" a term for more polished speech and end with "amar" a more discursive type of speech.