If you look at the "skills" menu in the game each type of armor has certain skills. For example I just started a tank who is wearing 5 pieces of heavy, one medium and one light. If you look under light armor skill there are perks there and some of those have to be a certain level before you can use them. So I I get my light armor skill high enough I can spend a point in let's say increasing magika recovery(I'm not at home so don't have access to the exact ones), which although great for sorcers comes in handy for my tank also. That's why you would want some in each as you'll get all the perks in heavy but there might be a few under the other two you would like also. If you don't wear either then that skill line won't even be visible.

I'll try to break this down.

For PVE, the different types of Armor have passives that substantially benefit different roles and builds. For Magicka Damage Dealers and Healers, light armor passives are ideal. For stamina users, the medium armor passives are beneficial for dealing for damage. For tanks of any variety, heavy armor is preferred for its greater protection. In PVE, you ought to be wearing the appropriate weight of armor for your build - if you need heavy armor for protection in PVE and you aren't a tank, you need to look at using food buffs, self-heals, and resistances so that you aren't completely squishy. Not using the appropriate weight of armor is a DPS/sustain loss, which is why your stamina or magicka Damage Dealers and Healers should not be wearing heavy armor.

For PVP, armor sets depend on your role. Heavy armor can be much more beneficial in PVP for the extra defense and procs, such as with 7th Legion. However, the more experienced you get, you can often run light or medium armor builds with a more glass cannon playstyle.

The reason people tell you to level all types of armor is two-fold.

1. If you later decide to swap your character around to fill a different role, you'll already have that armor line leveled up. My MagDK is a heavy armor PVE tank AND a light armor PVP healer, so I use both armor passives in different content. Later, I decided to run the Dark Brotherhood quest with that character, so I started leveling medium armor for the better sneak passives.
2. The Undaunted skill line has a passive that increased your max resources by % according to how many different weights of armor you are wearing. For some builds, its beneficial to wear 6/1 of 5/1/1 of different weights of armor to take advantage of the extra stats from Undaunted.

As macsmooth stated, have you unlocked the other armor lines on your sorc? You unlock them by equipping 5 pieces of that armor weight. I believe you still gain exp towards that weight from wearing the armor even if its not unlocked, IIRC.