What I've learned from experiments.
Weapon Damage
The level difference seems to have some impact. The high level character will not insta-kill most mobs on a low-level map. Evidence suggests that the weapon damage of the higher level character is scaled down. Specifically, I can visit old russia on earth with my level 20 character and still not one-shot everything. They go down easier than I remember, but some mobs may still require 2 shots.
Player Damage
It seems mobs also have a way of damaging you. If I run into a group of level 2 mobs on old russia as my level 20, I will still take damage enough to kill me. It seems to take longer than I remember but I'm not safe.
Experience
The level 1 character will earn XP if he's close enough (distance-wise) to the level 10 character. He will earn the same XP whether he kills the mob himself, or if it's killed by the level 10 character. Dregs gave us 20 xp each whether we helped kill it or not.
Specifically we tested with the lower level character just running alongside me and when I killed mobs, he got XP. Same way the other way around, if he killed a mob, I got XP even though I did nothing except tag along.
The same XP is awarded to both players, so obviously the higher level character will require a lot more dregs to be killed before levelling up.
Note the distance remark above. If the two players venture too far apart, no XP is rewarded from kills the other player does.
Item Drops
Item drops will mostly be according to the zone level. If you're playing as a level 10 character in a level 2 zone, you'll get level 2-3 drops, with the occasional item more appropriate for your level. Encrypted engrams will, as noted elsewhere on the interwebs, be decrypted at the level you have when you decrypt them so they're just as useful in a level 1 zone as a level 10 zone. No idea whether drop rate is affected though.
Item drops are also personal, which means that all the ammo and item bulbs that you see on the ground is yours.
Mission Rewards
Mission rewards is a one-time event. You only get the rewards when you complete the mission the first time. However, as a higher level character you have a higher chance to get encrypted or decoherent engrams containing shiny loot, which offsets this. It is not a waste of time to play with your lower level friends!
So to recap:
- Low-level character earns XP when he kills mobs, and he earns the same XP when the high-level character kills mobs, the latter only if the two characters are actually playing together (ie. not on opposite ends of the map). I don't know what the distance range is but it wasn't exactly stand-on-top-of-each-other either.
- Low-level character earns item drops as usual, according to the zone. As does the high-level character, but he'll get some item drops that aren't useful (too low level).
- High-level character is slightly overpowered for the zone, but not to the extent that you can level the zone just by sneezing.
Conclusion
It's probably best to be fairly close in levels, then both/all players will have the most fun and the most to gain, but there seems to be no adverse effects of bringing a high-level friend to help a low-level character, except for the high-level character not really getting a lot out of the process except the friendship.
Mega-conclusion
Destiny is meant to be played together, it seems Bungie has made a system that allows this, even if the players are not the same level.
So team up soldiers, the darkness must be pushed back!
Mission rewards at the end are based on the level of the mission. Increasing the level/difficulty of the mission will increase the level of the reward.
You can only get items with light stat from missions that are level 20+, strike playlist, raids, and the crucible. They will not appear in regular strikes or missions under level 20.
You are correct about the more enemies meaning more chances for engram drops, but you could also just farm explore/patrol mode for engrams instead of missions.
I'm pretty certain the highest rarity non-level 20 items only go up to blue quality, and blue engrams drop consistently from level 1 to level 20.
Best Answer
It does not affect the story, it merely sets your level to 25. It does unlock The Taken King Story Missions and and equips the character with level appropriate gear. Since the level requirement for the other missions seems to be below level 25, all those missions will unlock as well.
Note: You do NOT gain 25 levels of experience. You will be a "fresh" level 25 player, meaning you cannot allocate points into skills or anything.