You still only have a 5' range for Attacks of Opportunity (with the bow).
Attacks of Opportunity can only occur in squares that are Threatened by a creature.
You threaten all squares into which you can make a melee attack, even when it is not your turn.
This means that without Snap Shot (or some way of obtaining melee attacks while holding a ranged weapon) a creature holding a bow can't make Attacks of Opportunity at all*. Snap Shot not only gives your bow a Threatening range of 5', but also allows the bow to make those Attacks of Opportunity in the first place. Because you do not threaten squares while holding a bow, natural reach does not apply, you're only able to threaten at all due to the fact that you have Snap Shot.
So for your example, an unarmed Storm Giant is able to take Attacks of Opportunity with his weapons (or unarmed with Slam) at range 15', but once he wields his bow with both hands his Attacks of Opportunity with the bow would drop to a 5' range.
I don't think this is entirely unreasonable, as shooting a ranged weapon at close range is actually quite difficult! your target can increase the arc or their movements with much less effort, causing you to correct more. It also serves as an obvious balancing factor to the fact that ranged attackers are able to hit melee attacker long before the melee attacker get close the distance.
You can increase this range to 10' with the Improved Snap Shot feat
*There is an argument to be made that creatures with natural weapons (Slam, in this case) or PCs with Improved Unarmed Strike could take unarmed opportunity attacks with a free hand while wielding a 2-handed ranged weapon such as a bow. This is currently unresolved, so is 100% subject to GM ruling.
The actual question in the link above is only tangentially related to this question, but the core disconnect is relevant: Are bows considered 2-handed weapons, or are they something different? If you can remove your hand from the bow as a free action like you can with 2-handed melee weapons, then you're free to make Attacks of Opportunity, otherwise you're not