What is the nature of the prohibition to destroy bread? Why is it commonly presented differently from other destruction (bal tashchit)?
There appears to be an independent prohibition to throw out bread, and doing so could cause one to become poor (cf. Shabbos 143a, Tosfos to Brachos 52b) due to the respect and appreciation that we should have towards Hashem's sustenance that He has provided (cf. Aruch Hashulchan 180:4, Kaf HaHayyim 24:47-48). This rule-of not discarding bread with the rest of one's garbage-probably doesn't have to do with Val tashchis, because it applies only to pieces that are a kezayis or larger (Shulchan Aruch 180:4 though see Mishnah Berurah 180:10 for a stringency not to discard bread crumbs that would combine to be a kezayis).
Bread may be used for any purpose as long as it is not used in a disrespectful manner and the bread will not be ruined by what you are doing to it.7 One should not pass a cup of wine over a piece of bread because the wine may fall on the bread and make it uneatable. Similarly, one should not place raw meat onto a piece of bread.8 One is allowed to use bread to cover food in a utensil, since this is not showing disrespect to the bread because it will be eaten as part of the food.9 Accordingly, it is common for caterers to use a bread-like food to cover the soup at weddings to retain the heat.10 When reciting kiddush one may place a siddur on top of the challah cover.11 Similarly, one who wants to lift up a sefer may place bread underneath the sefer in order to lift it.12 When washing mayim achronim one should be careful not to let the water fall onto bread.13 Many caterers bake bread in a round shape, hollow the inside, and fill the space with soup (the bread is hard so no soup leaks through). It would seem that if the people eat the bread (bowl) afterwards this may be permitted.14
7 Mesechtas Berochos 50b, Tosfas “ein zurkin,” Rosh 7:32, Shulchan Aruch 171:1, Mishnah Berurah 171:2. See Biur Halacha 171 “lo.”
8 Refer to Rashi Mesechtas Berochos ibid “ein,” Rabbeinu Yona, Rif page 37b, Shulchan Aruch 171:1, Pri Megadim M.Z. 1, Mishnah Berurah 171:2, 5, Kaf Ha’chaim 2.
9 Pri Megadim M.Z. 171:2, Mishnah Berurah 7.
10 Horav Yisroel Belsky Shlita. Refer to Torah L’shma 106.
11 Birchos Hashem 3:17:footnote 62.
12 Eitz Hasadeh 17:footnote 8.
13 Kaf Ha’chaim 171:4.
14 Refer to Shulchan Aruch 171:3.
It would seem however that whether bread really has a unique status might be subject to a dispute amongst the rishonim. From http://halachayomit.co.il/EnglishDefault.asp?HalachaID=2821 :
Regarding bread, however, the Rishonim disagree: According to the Tosafot, even if the bread will not become ruined as a result of throwing it, such as by throwing from side to side on a table, it is still forbidden to do so, for bread has a special significance and throwing is considered degrading towards it. The Rosh, Tur, and other great Rishonim rule likewise. (The commentary Tziyun Le’Nefesh Chaya, ibid, indeed infers this from the language of the Baraita.)
On the other hand, the Rashba and Rabbeinu Yonah (quoted by the Bet Yosef in Chapter 171) write that there is no distinction between bread and other foods and as long as the food is not ruined by throwing it, it is permissible to do so.
As far as why we would treat bread differently from other foods (assuming that is indeed the case) - I assume it has to do with the religious significance of bread in halacha (as seen in it's roles in the avodah e.g. in the menachot, lechem hapanim, shtei halechem, etc.) and the fact that it is representative of food in general (which is presumably why the hamotzi blessing exempts the brachot of all other items eaten as part of the meal, i.e.: with the bread). Also, I believe there may be midrashic/kabbalistic analogies between man and bread.