Unusual iPhone problem

Hi there everybody, this is my first post as I am new to the forum. I have a problem with my iphone 4 that hopefully someone could tell me how to fix. It all started about 3 months ago when I dropped my iphone and accidentally bent the side silver frame. I kept using it until about a month ago when the screen would only illuminate an entirely black background, even though I could still receive texts and phone calls (though I couldn't answer them because my screen was not working). Because of this, I backed up everything on my phone to itunes on my computer and went to my local apple store to get my phone replaced. I was able to get a new iphone, and I plugged it into my computer as soon as I got home to get my precious apps back onto the phone. When I first plugged the phone into itunes, however, I updated my software first before restoring the phone (I had VERY old software in my original phone dating back to whatever was used in June 2011). After iTunes said the update was successful I noticed that the following was restored to my phone:
-Contacts
-Text messages
-Phone call history
-Notes
-Photos
but NONE of my music, NONE of my apps and none of my voice memos were (this was originally expected by me though, as I originally thought that EVERYTHING will be restored when I click the restore button on iTunes). What happened next is my problem. When I tried to restore the iPhone, a message came up saying that I had too little space on my phone for everything to be transferred back. I looked and saw that I had around 4 gb of space left. What confuses me is that somehow my iphone had less space even though nothing new was added to it (I didnt download any apps before trying to restore my new iphone). Can somebody please tell me how I can get my music and applications back to my iphone?