Does Barbarian's Greater Rage stack with the Rage spell?

Question

When a barbarian reaches 11th level they gain greater rage which increases the bonus that they would get when raging.

The spell rage

Each affected creature gains a +2 morale bonus to Strength and Constitution, a +1 morale bonus on Will saves, and a -2 penalty to AC. The effect is otherwise identical with a barbarian’s rage except that the subjects aren’t fatigued at the end of the rage.

Since the spell states that the effect is otherwise identical to the effects of rage, does that mean if a 11th+ barbarian has rage cast on them they instead apply the benefits of greater rage instead of the listed lower benefits which are lower than the ability rage?

Answer 1

No.

It seems like the question in your title and the question in your description are two different things, so I'll address them separately.

Does Greater Rage stack with rage?
No. A barbarian's Rage ability (and, by extension, Greater Rage) and the rage spell both provide morale bonuses.

Can a barbarian with Greater Rage who receives a rage spell apply the benefits of Greater Rage instead?
No. Since this is not specified, it isn't allowed. For comparison, a skald's Inspired Rage ability specifies exactly this-- if your Greater Rage-capable barbarian comes under the effects of a skald's Inspired Rage, you can apply your Greater Rage bonuses instead.

Answer 2

The spell says "The effect is otherwise identical..." which means that anything that isn't about the raw stats boosted (or the fatigue) applies. It's not Rage, but it pretends to be rage, but the specific numbers buffed aren't pretending to be rage - they're their own thing.

Now, if you had a Rage Power that gave you some additional nifty effect while in rage, then you could exploit that, but things that directly affect the rage bonus numbers look like they go nowhere.

So, no.