Does material change armor protection or just armor duration?

Armors in minecraft protects better as they are made in different materials or do they simply last longer?

Related to that, the more shirts on the bottom right, the better protected I am or does it simply reflect the armor's duration?

The reasoning is: If the material impacts only on duration then I'm better off keeping my iron for carts and rails and do a lot of cow's leather armor.

Solution 1:

Durability for armor is the amount of damage they can absorb when worn. From chand's answer:

             Base        Durability (½hearts absorbed)
             Protection  Leather  Gold  Iron  Diamond 
 Helmet             1.5       34    68   136      272
 Chestplate         4         49    96   192      384
 Leggings           3         46    92   184      368
 Boots              1.5       40    80   160      320

So, better armor certainly last longer. Do they provide better protection though? It depends.

The protection factor (% of damage absorbed) is:

   Sum of durabilities
 ─────────────────────── × Sum of base protection × 8%
 Sum of max durabilities

So a brand new full-body leather suit will give you 80% protection, just like a brand new diamond suit. However, material does make a difference in durability too — in both good and bad ways.

Example:

Armor       Material Durability Base P  Armor       Material Durability Base P
Helmet      Leather      17/ 34    1.5  Helmet      Leather      34/ 34    1.5
Chestplate  Leather      25/ 49    4    Chestplate  Leather      49/ 49    4
Leggings    Leather      23/ 46    3    Leggings    Leather      46/ 46    3
Boots       Leather      40/ 40    1.5  Boots       Leather      20/ 40    1.5
 ↓                      ─── ───   ────   ↓                      ─── ───   ────
Protection:  49% = 8% × 105/169 × 10    Protection:  70% = 8% × 149/169 × 10

Armor       Material Durability Base P  Armor       Material Durability Base P
Helmet      Leather      17/ 34    1.5  Helmet      Leather      34/ 34    1.5
Chestplate  Leather      25/ 49    4    Chestplate  Leather      49/ 49    4
Leggings    Leather      23/ 46    3    Leggings    Leather      46/ 46    3
Boots       Iron        160/160    1.5  Boots       Iron         80/160    1.5
 ↓                      ─── ───   ────   ↓                      ─── ───   ────
Protection:  62% = 8% × 225/289 × 10    Protection:  57% = 8% × 209/289 × 10

A half-worn leather suit with brand new iron boots (the cheapest iron armor) protects much more than a half-worn leather suit with brand new leather boots (62% vs 49%). However, a brand new leather suit with half worn iron boots protects much less than a brand new leather suit with half worn leather boots (57% vs 70%).

This is because the total armor wear is not the average of each item's wear, so the better material an item is, the more heavily it weighs in determining the overall quality. As shown, this makes things better and worse.

So, if you need a quick boost to your survivability, temporarily swap in iron armor; be aware that once it starts wearing down, those iron boots are going to be a liability, rather than a boost. To use them to their fullest you'll need a full iron suit.