How were these non-Spartiate members of the army regarded among the Spartiates on campaign with them?
From the time of the Battle of Thermopylae through the Peloponnesian War, what is known about the dress and armaments of the Perioikoi and Helot soldiers that would also make up the Spartan army? Could one distinguish between a Spartiate hoplite and a Perioikoi hoplite based on dress?
For Perioikoi: frustratingly little is actually known about them. Their name means "those dwelling around" and they are clearly occupying a physical and social space between the full Spartans, the Spartiates, and the Helots.
They were free, but were not citizens of Sparta. They were Lakedaimonians, or at least seem to get grouped in that term. They seemed to have their own shrines and temples, which suggests they were intentionally excluded from "Spartan" religious identity (that's a whole other question in itself, though; see Parker in Powell, Classical Sparta 145ff.
Herodotos says 5,000 perioikoi were present at Plataia in 479, and an equal number of full Spartiates (9.10ff). This suggests there were a lot of them, even at the end of the Archaic period, and we cannot expect that nearly all of the adult male perioikoi were in the army. We have no idea how or where they fought, but the assumption is that they were in the phalanx with the Spartiates. This would suggest they had the same hoplite equipment as the Spartiates. This in turn suggests that they were not poor themselves, and our limited evidence supports the notion that the perioikoi were at least partially "middle-class" or even wealthy. Xenophon famously calls an expedition of perioikoi soldiers in the early 4th century kaloi kagathoi, a term which is synonymous with the Greek elite (Hell. 5.3.9). I don't quite know what to make of that, myself.
If Herodotos can be trusted that there were 5,000 of each at Plataia, that seems to argue against the inclusion of perioikoi in the ranks as a stop-gap measure during the Pelop. War to preserve the numbers of full Spartiates.
That the perioikoi were very much trusted by the Spartiates seems pretty clear, even though some got up to rebellious mischief at various points. We hear of one being sent to scout out the strategic situation at Chios in 412, during the war, suggesting he was both trusted and also knowledgable of military matters. Another one commanded a ship (Thuc. 8.22.1).
Cartledge has a good modern discussion of the perioikoi, especially starting at p.153: Cartledge, Sparta and Lakonia. a regional history 1300–362 BC, Routledge, 1979.
More recently, see also:
A. Powell, Athens and Sparta. Constructing Greek Political and Social History from 478 BC. Routledge, 1988.
A. Powell, ed. Classical Sparta-Techniques Behind Her Success. Routledge, 1989.