I have always seen clerics wearing armor, and I have never seen them suffer from arcane spell failure. Yet when I check the rules I cant seem to find anything which allows them to not suffer these penalties. Have I somehow overlooked this or does the class granting armor proficiency grant it the ability to wear that armor and not suffer from ASF? It is just the fact that its arcane spell failure and not spell failure?
It specifically mentions that warlock eldritch blasts and invocations are subject to arcane spell failure because unlike most spell-like abilities they have somatic and/or verbal components, albeit simple ones that can be cast while wearing light armor (or medium with Battle Caster feat).
So in that situation, no they couldn't. And since spell-like abilities are in all other circumstances non-somatic/verbal, there's no Still/Silent Spell-Like Ability feat (unlike Empower/Quicken/Maximize). So you'll never be capable of casting them in that situation.
From Complete Arcane, page 7:
All invocations, including eldritch blast, have a somatic component.
Unlike other spell-like abilities, invocations are subject to arcane spell failure chance.
Here are the rules for Arcane Spell Failure:
Casting an Arcane Spell in Armor: A character who casts an arcane spell while wearing armor must usually make an arcane spell failure check. The number in the Arcane Spell Failure Chance column on Table: Armor and Shields is the percentage chance that the spell fails and is ruined. If the spell lacks a somatic component, however, it can be cast with no chance of arcane spell failure.
The rules do state a few exceptions (spells without somatic components in the quoted portion, and bards getting light armor without ASF elsewhere in the linked document), but proficiency is not one of them.
In other words: Even if you are proficient with armor, you still suffer its Arcane Spell Failure chance.
Even though it says that a magus can cast magus spells while wearing light armor without incurring the normal arcane spell failure chance?
Kensai is an archetype of Magus. The Magus does have a class feature that allows them to use light armor without spell failure:
Weapon and Armor Proficiency
A magus is proficient with all simple and martial weapons. A magus is also proficient with light armor. He can cast magus spells while wearing light armor without incurring the normal arcane spell failure chance. Like any other arcane spellcaster, a magus wearing medium armor, heavy armor, or a shield incurs a chance of arcane spell failure if the spell in question has a somatic component. A multiclass magus still incurs the normal arcane spell failure chance for arcane spells received from other classes.
But this is entirely replaced by the Kensai class feature of the same name:
Weapon and Armor Proficiency
A kensai is proficient in simple weapons and in a single martial or exotic melee weapon of his choice. A kensai is not proficient with armor or shields and suffers normal arcane spell failure chance when casting magus spells while armored.
In other words, the class behaves as though the entire "Weapon and Armor Proficiency" block had been removed from the Magus, and replaced with the one from Kensai.
The Kensai will also not get the Medium Armor (Ex) or Heavy Armor (Ex) class features of the Magus class, as these are replaced by the Iajutsu (Ex), and Iajutsu Focus (Ex) class features.