Philo stone isn't a good item in my opinion anymore. Here is why:
1) The regenerations stats have been nerfed and are too low to be used for jungle sustain.
2) With the regen being nerfed, the g/10 is the only really attractive thing about the item, but the only way to gain utility from it is to not upgrade it till late game. However, you actually want the upgrade prior to late game to enhance your ganks...
The Spirit Stone gives good regeneration/sustain statistics for the gold spent. Effectively it's just the old Philosopher's stone (season 2, you can look it up), but without the G/10.
You are correct in thinking that Spirit Stone is a better item, because it just is, especially on amumu.
One thing to also consider is the fact that the item sightstone exists. This is, in effect, a gold per second item because you can place a lot of wards without having to pay for them after the initial purchase of the item. This item also costs just as much as the philosopher's stone.
If I were you I would recommend buying your spirit stone once you go back to base, then save up for sight stone or the lucky pick item. I do not jungle amumu, however I am familiar with his ganks and mechanics. The one problem amumu has right now is that his pre-6 ganks are somewhat weak if he has no follow up CC from his laner. Having Twin Shadows as an item allows him to do the following when ganking:
1) Slow down opponent and then stun with his Q.
2) Stun and then slow with the item's unique activation.
Try to look at current season three specific guides for junglers as a lot of the older, season 2, guides are out of date.