The skills have different primary effects, though there is some overlap. In particular:
The spellbook skills (Wizardry, Divinity, Alchemy, and Psionics) are important for determining whether you can learn a spell in the first place. In particular, your spellbook skill + 10% of your realm skill is used to determine if your skills are high enough (15 for 2nd level spells, 90 for 7th). Note that spellbook skills are harder to raise, especially since a Bishop casting a multi-school spell (like Heal Wounds) only gets skill experience for the highest skill (exception: if silenced, only Alchemy gets skill experience). In general, particularly for a Bishop, you should favor putting points in these skills (but since the first few points of any skill are hard to earn, you might want to put at least 1 or 2 points into your realms).
The realm skills affect how reliable your spells are when cast at higher power levels, as well as how many spell points you get of the realm. Spellbook skills do have a minor effect, but realm skills are more important for these purposes. They are easier to increase through use than spellbook skills.
Note that level also matters; learning higher level spells requires that you reach the required minimum level as well as having good skill, and your higher level spells won't be safely usable at high power levels no matter how high your skill is unless your level is 6 more than the minimum. (Also, for 7th level spells, I believe 100 spellbook + 100 realm isn't quite enough to use them 100% reliably at power level 7, even at level 24.)
There's also Powercast, which is earned by raising base Intelligence to 100 and which works completely differently; it increases the power of your spells (effect varies by spell) and their ability to penetrate magic resistance, but does not affect learning spells or whether the spell fizzles or backfires.
