http://www.gamesas.com/topic/1378689-bows-and-followers/
Scroll down a bit of ways til you see a makeshift chart.
Basically they can't use anything beyond a certain "max damage" depending on bow/arrow type used.
Edit: Bah nevermind I'll just qoute it.
I enchanted whole bunch of bows with absorb and fire so that it's visible when the follower uses it. I think Eord is right about his observation. It is consistent with ChefDom's finding but more refined.
When I gave Lydia low level arrow like iron arrow or forsworn arrow to use with the bow, she used up to Glass Bow. When I gave Lydia Daedric arrow, she only used the Long Bow and nothing stronger.
Roughly, the follower will use the bow when the base damage value of the bow + arrow is between 15 and 22. Not to strong, not too weak.
Here is the data I collected:
Long Bow 6
Hunting Bow 7
Imperial Bow 8
Dwarven Bow 10
Elven Bow 12
Glass Bow 14
Daedric Bow 18
With Forsworn Arrow (7 damage) 7 + 7 = 14
Long Bow no 6 + 7 = 13
Hunting no
Imperial yes 8 + 7 = 15
Dwarven yes 10 + 7 = 17
Elven yes 12 + 7 = 19
Glass yes 14 + 7 = 22
Daedric no 18 + 7 = 25
With Iron Arrow (8 damage) 7 + 8 = 15
Long Bow no 6 + 8 = 14
Hunting no
Imperial yes 8 + 8 = 16
Dwarven yes 10 + 8 = 18
Elven yes 12 + 8 = 20
Glass yes 14 + 8 = 22
Daedric no 18 + 8 = 26
With Ancient Nord Arrow (10 damage) 7 + 10 = 17
Long Bow yes 6 + 10 = 16
Hunting no
Imperial yes 8 + 10 = 18
Dwarven yes 10 + 10 = 20
Elven no 12 + 10 = 22
Glass no
With Orcish Arrow (12 damage)
Long Bow yes 6 + 12 = 18
Hunting no
Imperial yes 8 + 12 = 20
Dwarven yes 10 + 12 = 22
Elven no
With Dwarven Arrow (14 damage)
Long Bow yes 6 + 14 = 20
Hunting no
Imperial yes 8 + 14 = 22
Dwarven no
With Elven Arrow (16 damage)
Long Bow yes
Hunting no
Imperial yes
Dwarven no
With Glass Arrow (18 damage)
Long Bow yes
Hunting no
Imperial yes
Dwarven no
With Ebony Arrow (20 damage)
Long Bow yes
Hunting no
Imperial yes
Dwarven no
With Daedric Arrow (24 damage)
Long Bow yes 6 + 24 = 32
Hunting no
Imperial no