Aside from having armor runes on a jungler I have not seen much use for getting specific rune pages. I reorganize mine every once in a while and I cant' seem to feel a difference in gameplay.

Where do runes normally help and what are the benefits of making pages specific to champions?

they help at early levels. for example, corki starts with 48 AD, and a normal rune page for him will give an extra 15 AD - that's a 30% increase to all of his auto attacks.

There are three cases in which they matter:

1) You need runes to jungle effectively. Not just Armor runes, but also something like Attack Speed or AP to make your clear time not terrible.
2) You need runes to support effectively. gp10 yellows and quints are required in order to get your gp10's at any kind of decent time, otherwise you'll just spend *all* your money on wards and just fail late-game.
3) You can really boost your early-game with a smart rune setup. Flat runes have a pretty significant effect at low levels.

Beyond that, they're not incredibly important, but still give an edge. That's why you see people play scaling runes, to get an edge late-game over people who got flat runes early-game.

All that said, you shouldn't bother with runes until level 20. Tier 1 and 2 runes are really just too small to matter, especially since you don't have any notable number of rune slots, and should be ignored. Once you get Tier 3 runes, though, then IMO you're good to go for normals against 30's, because those last 10 runes really aren't that huge.

Yes. Runes and masteries can give you around 2-3 starting item's worth of stats.

Another thing that runes can help with is last hitting: Even a few points of additional damage give you more leeway for the "one hit from dead" window of opportunity.

They make a difference.

Runes usually help early-game, when harassing is key to bullying opponents out of lane and claiming free minion farm, and occasionally late-game if you have per level runes (these are usually used to make up for a lack of item purchases late-game - i.e. beefing yourself up with magic resist per level runes if you don't usually buy a lot of magic resist on your assassin or nuker characters).

While all champions benefit from having specific rune pages, some do so more than others. Bruisers like Jax, Darius, and Lee Sin can usually get away with similar rune pages (armor, attack damage, magic resist per level) that allow them to trade well early-game and late-game give them a bit more beef against nukers that can easily burst them before they get damage out.

By contrast, Akali is a champion who benefits IMMENSELY from her own rune page - so much so that some players will swear that the only way to play Akali is to spend 100+ games farming up the perfect rune page for her. This is mainly due to her passive, which requires she have enough starting AD and AP to activate, having AP and AD runes benefit her immensely early-game. Not to mention that she usually doesn't build any AD items except her Hextech Gunblade (which is expensive and denies her the AD half of her passive for the early-game), so if you want both her AP and AD passive up you're best off just making a rune page for her. She is worth it (very strong top lane and ganker) but it takes time.

I like to harass early and try to score first blood, or any kill as soon as possible. So I use flat runes, it gives me a significant boost. But as already mentioned, don't waste your IPs in tier 1 and 2 runes, they really don't worth it.

This is very true.

I'd consider myself very experienced when it comes to runes; I have over 10 sets of all four types. That said, rune pages can make a big difference in-game - but not always. The right rune combination paired with the right masteries on the right champion WILL decide a game. Take Garen for example: 3 Flat Attack Damage quintessences, 9 Flat Attack Damage marks, 9 Flat Armor seals, 8 Magic Resist per Level Glyphs, and 1 Flat Attack Damage glyph. With this page, you'll start with 16 AD, and 13 Armor. This makes a HUGE difference when added to a Dorans Blade and masteries; 91 AD at level one is devastating.

That 1 Flat AD glyph is silly. The game only round the number to display it to the user, it doesn't do it for the damage calculations.
Some use 2 MPen glyphs because this, combined with Marks and Sorc boots, will add up to 30(?) making it penetrate all of the base MR of a lot of champions. Getting flat AD glyphs makes no sense though.