Where are my old apps? Not on new phone.

Question

I tried transferring my 4s to my 5s with icloud and didn't move 90% of my apps. I tried it again backing up to the computer from 4s then restoring from computer to 5s, again 90% are missing, only the same few apps moved over besides the new ones that came with 5s.  Now if I look at either phone in itunes, all my apps are pictured on the right side (under apps) but nothing is listed on the left side where they all used to be listed. Same with the 5s, the few apps are listed on the right side but there is nothing listed under Apps on the left side.
Where are all the apps I bought--practically all where bought from the Apple app store. How can I get them and put them on my 5s? Is it because they aren't 64bit? I thought the 5s could handle both. I really need these apps.
Help!

Answer 1

Thanks Steve for that quick response!  From other unrelated comments I've read, Apple never helps with password problems, always says, "We don't keep your passwords, so there's no way we can help you with passwords."  But since this is actually more related to PURCHASED-apps not appearing in iTunes, then maybe they would have a more helpful attitude regarding that.
However, the joke is on me, as I think I was taking gail-from-maine's sentence too literally where she wrote, "
Sign into iTunes with the Apple ID that you purchased the apps with."  She probably meant different family members might have different IDs/different phones, so make sure you're using the correct one for whichever phone.
Because I went ahead & used my CURRENT ID/PIN, & followed her directions so far, & yes, whew!, there are 222 apps showing in iTunes.  However, they ALL show the little cloud on them, which gail said means they are NOT on the Mac.  But they ARE on the Mac, under User > Music > iTunes > Mobile Applications.  So I'm cornfused.
I also notice that some of the apps (in the long list under Mobile Applications/Mac) show a "Creation Date" of 12/31/69 (!!) so I don't know how/why it says that as my clock/time/date is correct/current. Might that be causing a problem?  There appears to be no way to change that "Creation Date" in "Get Info."
Another question, what happened to the way you could move/re-arrange the location of the apps in iTunes according to screen/pages on the phone?  (with OSX Lion & iTunes 11.1.5, it simply shows all the apps in iTunes in 27 rows of 8 apps each row, but I see no way -- YET -- to rearrange them by different phone-screens/locations, the way it used to show them in older iTunes & you could move them around to where you wanted them to show up on your phone).
FWIW, I did try doing as gail-from-maine said, "Click on them [in iTunes] to download them [the ones with the clouds on them] to the Mac," (in the lower right of the iTunes page, it has the option/button to "Download ALL"), but it didn't do anything, just a quick blink like flutter/action on iTunes page & no indication that anything was downloading.
I'd rather have the copies that are already on the Mac under Mobile Applications put on the phone, since they are the ones with CONTENT in them through the last backup, right?  (I don't even know if the 222 apps showing in iTunes actually have any CONTENT in them???)
As you can tell, I'm not an experienced iTunes user, never cared for that software much.