I have found that Tekkit is the most amazing Minecraft mod ever, but I notice that often Technic is also listed with Tekkit. So, I must ask: What is Technic and how do I use it (is it a server mod, client mod, etc.)?
I'd say that the most major difference is the addition of the Thaumcraft and Mystcraft mods in Technic. As others have said, Tekkit is for use in a multiplayer setting, while Technic is for SSP only.
The major mods in both are IndustrialCraft2 (adds electricity and industrial machines), Equivalent Exchange 2 (adds in alchemy using the Philosopher's Stone, and several highly powerful end-game tools like the Void Ring, Ring of Arcana, etc), Redpower 2 (adds in redstone wiring and machines), and Buildcraft (adds in some automated mining machines as well as transport tubes and controls for that).
In addition to those core mods shared by both packs, Technic adds several features that are best in singleplayer. Major differences are:
A few other minor changes worth mentioning:
Tekkit is for multiplayer.
Mods in Tekkit:
Technic is for singleplayer only.
Mods in Technic are as the ones above, plus:
As you see there is a lot more mods in Technic.
While Tekkit is an SMP mod, Technic is SSP only. This means it cannot be played with others, but it allowed them to add quite a few mods which were not feasible in an SMP setting. I'd post a list of the mod differences, but the Technic site is blocked here, so I'll have to do so when I get home, unless someone feels like editing it in.
The easiest way to use Technic is to download the Technic launcher from their website. This will give you a choice to run Tekkit, Technic, Hack/Mine, Yogbox, or Voxel Modpack.