Is there a substantive difference in meaning when the Torah uses the Hebrew verb root דבר rather than אמר, or do they both pretty much mean "to say" or "to speak"? Do any sages see the choice of one word over another to have some extra meaning?
Many people tried to differentiate between "אמר" and "דבר".
There are several midrashim and a gemera in Makkos (11a) that identify דיבור as a "harsh" (קשה) way of speaking based on Gen. 42:30, and אמירה is "softer" (רכה). While some (including Torah Temima) take this literally to mean that דיבור and אמירה are different primarily in tone, others (N. H. Wessely, followed by Malbim) take this to mean a difference in style, where דיבור is much more elaborate and אמירה is is more get-to-the-point.
In HaPardes (No. 47, Vol. 9, pp. 24-25), Rabbi T. A. Sanders identifies some other qualities specific to "אמר" or "דבר". For example, דיבור is much more broad while אמירה is much more specific (similar to Wessely's distinction). אמירה is more impersonal, conveying a general sense of imparting information; it can be used for something said through a messenger or in writing; it can be used for something said to one's self or even in one's mind without actual verbalization. דיבור, on the other hand, is only used when there is a listener as well as a speaker.
The Midrash Rabbah (תשא פ' מב ופ' סח) and the Midrash Tanchuma (פר' צו) and the Sifri (פר' בהעלותך) teach that "דיבור" is a harsh way of speaking, and "אמירה" is a soft way of speaking.
The Vilna Gaon in Aderes Eliyahu on the first verse in Ha'azinu says that וידבר is linked to Torah Shebichsav, and ויאמר is linked to Torah Sheb'al Peh. He explains that this is why communication with Moshe is always with וידבר, and the only place where Hashem speaks to a Navi with וידבר is in reviewing a law from the Torah, when it reviews the laws of cities of refuge to Yehoshua (20:1).
The idea seems to be that דבר is a more direct communication than אמר (as the prophecy of Moshe was more direct than that of the Nevi'im after him).
R. Avraham b. haRambam writes in the beginnining of Va'Eirah (7:8):
לא ידעתי [מה החילוק] בין ויאמר לוידבר ולמה אמר פעם ויאמר ופעם וידבר ואם רק הוא מכם רק
"I do not know what's the difference between VaYomer and VaYedaber, and why sometimes the Torah uses one, and sometimes the other, and if it seems meaningless, it's your shortcoming."
Just a thought. On the words of דַּבֵּר ׀ אֶל־בְּנֵי יִשְׂרָאֵל וְאָמַרְתָּ אֲלֵהֶם וְעָשׂוּ לָהֶם צִיצִת where Moshe is instructed to teach the commandment of tzitzis, both words are used in this pasuk.
As mentioned above, the Talmud Makkos (11a), Midrash Rabbah (תשא פ' מב ופ' סח) and the Midrash Tanchuma (פר' צו) and the Sifri (פר' בהעלותך) teach that "דיבור" is a harsh way of speaking, and "אמירה" is a soft way of speaking.
Based on this understanding, one could possibly say as follows: When conveying the Halachos of something, that are by their very nature rigid and strict, one could do so in a strict stern manner, like a drill sergeant in the army, or in a soft way like a caring parent would do for their child. The Torah is perhaps teaching us here that even strict rules need not be taught with strictness, but in a manner that portrays the Torah’s value of kindness.
Based on Rashi, the general rule of scriptural interpretation is אֲמִירָה לְשׁוֹן רַכָּה. This means that words such as וַיֹּאמֶר denote gentleness. [See Rashi Shemos 19:3; Makkot 11a; Cf. Ben Yehoyada on Makkot 10b; Kli Yakar on Shemos 26:1] Rabbeinu Bachya says that this can be derived from the word אמרא, the edge of a cloak a person wears on his left side. [Similar to a paludamentum]
A verse that illustrates this symbolic idea of where G’d is depicted as wrapped in such a cloak, i.e. “עֹטֶה־אוֹר כַּשַּׂלְמָה - [You] enwrap Yourself with light like a garment.” [Tehillim 104:2]. This represents מִידַּת הַדִּין - the attribute of Justice. (The act of wrapping it and covering it over the left side symbolizes the covering up of the attribute of Justice, which is on the left.)
On the other hand, דִיבּוּר לְשׁוֹן קָשֶׁה - words such as וְַידַבֵּר, denote harshness. This can be derived from the word דְבוֹרָה - a bee. Although providing honey, i.e. sweetness, it also harbors a deadly sting. Nevertheless, the bee symbolizes the attribute of G-d that combines both the attribute of Mercy and the attribute of Justice. [Cf. Rabbeinu Bachya, Shemos Bo 13:1:4]
Since you mention the sages (or do you mean the Talmud?) wouldn´t it be better to consider the original root meaning (as indicated by their family usage in syntax) instead of trying to force biblical hebrew into modern mentality and usage? אמר is "be precise", whereas דבר is "arrange, set in order". Their distinctive meaning and use is clear if you think like the original hebrew authors and readers. That's why the idiom דבר לאומר (he spoke, saying) occurs hundreds of times indicating a direct verbatim quote. A literalistic translation from the original root meaning would be "he arranged, to be precise, thusly:" or to put it in our modern mindset: "he said, and I quote, '...'".