Okay, I'll have a look at what I've done. I think the problem may be something to do with me using a different version of forge than in the client - I used the installer on the client to installl it, but just direct-downloaded the .jar for Tekkit.
EDIT 1: I've looked at the downloads and realised that yes, I was using the next-oldest version because I went to the legacy links. My bad. I downloaded the latest one, shoved it in my installed-via-the-launcher files to see if it would work, and it does.
However, I was looking through the log after it failed to connect to my server, and I noticed that there were loads of errors. I think I'll just re-do the client-side pack from scratch, and see if that fixes it. Basically, it wouldn't connect to the server due to ID mismatches. All of them in the 20,000 to 21,000 range, if I remember correctly. Because there were so many I decided it must have been due to doing it all in Vanilla Minecraft's client at the start. I'll move the .jars and .zips again and see if that works.
Oh, and about the .zips... Forge seems to have trouble recognising them as mods. It does, but in the log it doesn't seem to like them. However, the mod instructions say just to drag them in and not extract them or anything.