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 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."