Would his equipment be different depending on his social status?

Like many periods in history when we speak of military equipment such as arms or armour we are inevitably discussing those who can afford it or who have been given it as part of a specialised, honoured band of professional warriors. The average levy, raised from the vassal city-states of the Triple Alliance, would just be average Joes in their loincloths bringing whatever weapons they had, such as flint-tipped or wooden spears hardened over fires.

Soldiers who demonstrated great valour and skill in battle (particularly by capturing potential sacrifices) were rewarded in several ways. Off the battlefield they were rewarded with fancy animal skins and given the rights to "drink the [alcoholic] beverage pulque (pul-ke) in public places, to dine in the royal palaces and to keep concubines"^1

On the battlefield, some of these skilled warriors would be invited to join the warrior-houses, such as the cuauhmeh (kwaw-meh) - the Eagle Warriors - or the ocelomeh (o-se-lo-meh (singular ocelotl or 'o-se-lotl, whence our word "ocelot")) - the Jaguar Warriors. Less well-known to popular culture are the other warrior houses, such as the otomih or Otomi Warriors, named for the Otomi people whom the Aztecs regarded as fierce warriors, and the cuachicqueh (kwa-chik-keh) or Shorn Warriors, whom Hassig mention are the Alliance's pride and joy, camping and marching (in and out of battle) ahead of the rest of the army and always being the first to engage the troops (but he's the only source I can find on this).

You had to earn your entrance into these prestigious warrior-houses. In return you could wear certain facepaints, feathers, headdresses and skins (i.e. Jaguar warriors wore Jaguar hides) which terrified the enemy and inspired your fellow warriors. They would wear costumes over their cotton armour (see below) called tlahuiztli (tla-wis-tli). These provided a small amount of additional protection, especially to the arms and legs, but are mainly for show.

How would a warrior be equipped going into war?

For the lucky or high-born deserving of special equipment, warriors would wear thick cotton "onesies" (for lack of a better term) known as ichcahuipilli (ich-ka-wi-pi-li) made from cloth stuffed with unspun cotton about two fingers in thickness.

For ranged combat they would use bows, throw stones, or employ javelin-throwers known as atlatl (two syllables - a-tlatl, not "addle-addle") which were small pieces of wood which provided a lever to launch darts further and harder than just by chucking them

Once melee was met, the most characteristic and popular Aztec weapon comes into play - the maquahuitl (ma-kwa-witl). Some people call it the Aztec Sword, others a "club". It would be easier to think of it as a cricket bat with sharp shards of obsidian, a volcanic glass, embedded or glued to the sides. Obsidian is sharper than steel but flakes easily and loses its edge. The Spanish made a few accounts mentioning that these razor-bats were able to behead horses in the heat of combat.

Warriors would be armed with circular shields known as yaochimalli made from hide, wood or thatched bamboo decorated with feathers.

this image illustrates a number of pieces of equipment I've mentioned. I can't speak to the accuracy of the designs or the depiction of the jaguar warrior (it was my understanding they wore actual hides, not a patterned costume) but it serves a purpose here.

The Aztecs also had spears, clubs and studded maces, the latter of which brings us to...

Were special weapons used for Flower Wars?

Not just weapons - the entire mindset of the army and individuals was different from how we understand it. The crippling and incapacitation of enemies for the purpose of capture was important not just on a philosophical scale (for the continued fuelling of the sun) but also immensely important as a ticket for social mobility for the combatants. Soldiers would aim to incapacitate and, as capturing someone as part of a group was far less impressive than doing it by yourself, combat was often a series of individual duels. Clubs with knobs or stone insets were used to break the limbs and knock out opponents, rather than hacking, cutting or causing them to bleed out and die before they could be brought back to the altepetl (the city) for sacrifice.

^1 The Illustrated Encyclopaedia of the Aztec & Maya, p. 87 & The Aztecs

SOURCES

Charles Phillips, The Illustrated Encyclopaedia of the Aztec & Maya (Anness Publishing, 2012)

Manuel Aguilar Moreno, Handbook to Life in the Aztec World (Oxford University Press, 2006)

Ross Hassig, Aztec Warfare: Imperial Expansion and Political Control (University of Oklahoma Press, 1988)

Barry L. Isaac, Aztec Warfare: Goals and Battlefield Comportment (Ethnology, Vol. 22, No. 2 (Apr., 1983), pp. 121-131)

Michael E. Smith, The Aztecs (Third Edition, Wiley-Blackwell, 2012)