Bow Pros:

1: Unquestionably stronger against unarmored/very lightly armored opponents, even without considering perks.

2: Overall better perk synergy, including: Berserker(3 shots!), Overwhelm, and Footwork and Rotation.

3: 8 vs 6 range (with perk). Significant against Archers.

4: Less "wasted" damage/easier to finish off low health enemies.

Crossbow Pros:

1: With perk, is quite a bit stronger against armored opponents (headshots are especially deadly, as most players know!).

2: Can wear heavier armor, including helmets (requires less fatigue to shoot).

3: More Accurate (most noticeable early on).

Overall:

Bows are stronger against:

Goblins, Wearwolves, Nachzehrers, Undead/Fleshy Dead (especially Necromancers!), Archers & low-medium armored brigands, human two-handers, Orc Berserkers. Also, are not as effective for low level archers.

Crossbows are stronger against:

Ancient Undead, Armored Humans/Nobles, Orcs (Warriors/Warlords), Fallen Heroes. Are more effective for untrained, low level archers.

In my opinion, I value Bows quite a bit more than Crossbows because Crossbows largely fill a niche that can be filled by your frontline brothers (2h Axes/Hammers, Duelists), whereas Archers, Goblins, and Necromancers can be difficult for your frontline to reach (Goblins especially).