Great that you finally made it! Not a day to late though as iPad have been around since...can't remember! 😉
 
But...Why on earth have you not at least kept all the features from the iPhone-version???
 
(I might have missed this but then it is not intuitive enough.)
 

Volume slider (this is the weird one) --- YOU FINALLY MADE IT!!! (Saw this 2013 Sept 25!!)
Be able to "drill down" on an artist "Now playing" (from a playlist, full view) and from there explore the albums etc. (#1)
Airplay menu in overview mode
Multi gestures (two fingers up or down for volume control)
To be able to search in a playlist (from jespersthlm)
Double tap the fullscreen should brings up the artist overview (same feature like the second from above)
As you have choosed to use the leaf navigation (sliding pages sideways for the hiearchy), dim down the pages in the background so it does not get so messy as everything have the same light grey background color now. 
NEW: When you do a offline syncing, the symbol appear in the left side lower corner. It would be great by clicking that symbol takes you to the place where the syncing is happening.
More features requested from users below:
 

History
Spotify Apps
Jump from song directly to Album
Improved pane navigation
Radio
Customizable tabs
Genres and Top Charts
Easy access to album and artist panes from Now Playing Menu (e.g. like in the desktop app where you click on the artist name) (#3)
Being able to "star" a whole album. (#2)
Swiping through cover arts in the Now Playing panes only starts playing a song when you *let go* of the swype, i.e. to allow you to skip over tracks easily.
View the queue list
 
 
 
SOLUTIONS:
 
#1 = Tap the full view screen and in the bottom you have a symbol with a bunch of people. That is the one!


#2 = (from user: soho/Spotify-team) It is actually possible to star a whole album. On the album view, just tap the + button and then select Starred in the "Add to Playlist" popover that appears.
 
#3 = Tapping the artist name in Now Playing will take you to the playing context. So if you're playing from an album, that's where you'll end up. If you're playing from an artist or a playlist, then you will be taken to the artist view or the playlist view. (i.e. same as on desktop)
 
COMMENTS:
 

(from user: soho/Spotify-team) We've had a lot of discussions about the context menus you get when long-pressing a track, tapping the small cover art in Now Playing Bar, and also seen at the bottom of the Now Playing view. We wanted to keep them clean and not too confusing, but the discussion is still ongoing and nothing is set in stone, so feedback is always welcome.