Said in one breath
These aren't the only changes between the Exodus set and the Deuteronomy set; a few other changes in other commandments are made as well. Shevuos 20b focuses on the changes in the fourth and ninth commandments, but the why should extend just as much to the tenth.
זכור ושמור בדיבור אחד נאמרו מה שאין יכול הפה לדבר ומה שאין האוזן יכול לשמוע
"Remember" and "Guard" [the Sabbath] were said in one word - that which the [human] mouth cannot speak nor that the ear can understand.
While this statement is said only regarding the fourth commandment, the Gemara explicitly connects it to the ninth as well:
בשלמא התם בדיבור אחד נאמרו כדרב אדא בר אהבה [...] אלא הכא למאי הלכתא מיבעי ליה
It's well there that they were said in one word - like [the statement] of Rav Ada bar Ahavah [who taught a halacha based on connecting "remember" with "guard"]. But here [by the exchange of "vain" witnesses for "false" witnesses in the ninth commandment], for what halacha do we need this? [The Gemara proceeds to deduce several halachos from this connection, which are beyond the scope of this answer.]
This statement therefore implies that whenever the Deuteronomy verses exchange anything with respect to the Exodus ones:
- The two terms were both said at Mount Sinai in the same breath.
- The exchange comes to teach a unique halacha by connecting the two principles.
So, returning to your question: part of the significance of the change is that it indicates that the two terms were said simultaneously on Mount Sinai, but furthermore, it indicates a connection between the laws of coveting with the laws of desiring. What is this connection?
What type of desire is mentioned here?
Shevuos is silent on the tenth commandment, but other sources aren't. Mechilta of R' Shimon 20:14 writes:
לא תחמד ולהלן הוא אומר (דברים ה' י"ח) לא תתאוה לחייב על תאוה בפני עצמה וחמדה בפני עצמה מניין התאוה אדם סופו לחמוד שנ' לא תתאוה ולא תחמוד:
"Do not covet." And there it says, "Do not desire" - to obligate on the desire separately and the coveting separately. How do I know that desire leads a person to coveting? As it says, "Do not desire," "Do not covet."
The way the Rambam understands this Midrash is as follows (Gezeilah 1:10,12; nearly identical wording is found in Shulchan Aruch, CM 359:10-12):
כֵּיוָן שֶׁחָשַׁב בְּלִבּוֹ הֵיאַךְ יִקְנֶה דָּבָר זֶה וְנִפְתָּה בְּלִבּוֹ בַּדָּבָר עָבַר בְּלֹא תַּעֲשֶׂה שֶׁנֶּאֱמַר "לֹא תִתְאַוֶּה" וְאֵין תַּאֲוָה אֶלָּא בַּלֵּב בִּלְבַד הַתַּאֲוָה מְבִיאָה לִידֵי חִמּוּד וְהַחִמּוּד מֵבִיא לִידֵי גֵּזֶל. [...] הָא לָמַדְתָּ שֶׁהַמִּתְאַוֶּה עוֹבֵר בְּלָאו אֶחָד וְהַקּוֹנֶה דָּבָר שֶׁהִתְאַוָּה בְּהֶפְצֵר שֶׁהִפְצִיר בַּבְּעָלִים אוֹ בְּבַקָּשָׁה מֵהֶן עוֹבֵר בִּשְׁנֵי לָאוִין. לְכָךְ נֶאֱמַר "לֹא תַחְמֹד" וְ"לֹא תִתְאַוֶּה". וְאִם גָּזַל עָבַר בִּשְׁלֹשָׁה לָאוִין:
Once he thinks in his heart how he can acquire this object and his heart is seduced in this matter, he violates a prohibition, as it says, "Do not desire," and desire only refers to in the heart alone. Desire leads to coveting, and coveting leads to theft [...] You have learned that one who desires violates one prohibition and one who acquires the object which he desires by pressing the owners or by requesting of them violates two prohibitons. Therefore it says, "Do not covet" and "do not desire." If he stole it he violates three prohibitions.
In other words:
- Desire does not just refer to desire, but also actively plotting how to get the object. (As an aside: The Torah never prohibits random thoughts which may pop into your mind; it prohibits dwelling on them. This is the point made by the Sefer HaChinuch 416, who notes that only thoughts which are in one's control are prohibited.)
- Desiring is taught in the same breath as coveting to teach that desiring indicates the type of desiring which can lead to coveting.
Coveting and desiring what?
That sufficiently explains why the verbs are swapped, but what about the objects? As you note, in Exodus we're proscribed from coveting houses, wives, slaves, animals, and "anything else," while in Deuteronomy coveting is limited to wives and desiring is applied to houses, fields, slaves, animals, and "anything else." These lists are identical with two important changes: wives aren't singled out by desiring, nor are fields by coveting.
Let me ask you a more basic question: If the verses are just going to say "and anything which is to your friend," why bother listing specifics? The Midrash (Mechilta ad. loc.) understands each point to teach that the laws of desiring and coveting are expanded to include more things. I won't go into the entire Midrash here, as the first line is sufficient to answer your question.
יכול לא יתאוה על בתו ליטלה ת"ל אשת רעך מה אשת רעך שהיא אסורה לך אף כל דבר שאסור לך:
I might have thought he can't desire his daughter to take her; the verse teaches, "Your friend's wife" - just as your friend's wife is forbidden to you, so, too, anything which is forbidden to you.
As noted earlier, your friend's wife is not mentioned at all by desiring, yet that's the verb the Midrash uses here. Seemingly, based on the above connection between the two, it's sufficient to make the one change by each and infer that the differences mentioned by one are applicable to the other.
- The changes are meant to signify that each variant was said simultaneously on Mount Sinai.
- The changes are meant to signify that the laws of one can be extended to the other; specifically:
- The switch from "covet" to "desire" teaches us which type of desire is meant: not just any desire, but specifically desire which will lead to coveting.
- Mentioning wife only by coveting and field by desiring allows us to extend the inclusions of each one to the other prohibition as well.