Robert Schumann (1810-1856)
The World's Greatest Composer

The symphony which we know as No. 4 was, in fact, the second Schumann composed, in the summer of 1841 soon after the success of the Spring Symphony (No.1).  It was premiered in December 1841, but was much less well received than its predecessor, maybe partly because of its unorthodox form and scoring.  All four movements are played without a break, there is a remarkable bridging passage between the third and fourth movements, and in its original version there was some strange instrumentation, for example a guitar in the second movement. It was unpublished and unloved until 1851.
At this point, Norbert Burgmuller's Symphony No. 2 comes into the picture.  As dealt with on the Burgmuller page, Schumann had worked on this.  It reminded him of his own D minor symphony, and only 10 days later he set about revising it.  The revised version is that which we almost always hear today, although its popularity had to wait until the day when its unconventionality would be better understood.  Today, the 4th Symphony is as frequently played and recorded as the 1st; we no longer consider conformity to be essential. It is my favourite Schumann symphony, second only to the Violin Concerto as my favourite work of all.
There are several complete performances of the 4th Symphony on YouTube, including an appalling rendering by Herbert Von Karajan (I wonder if he just wanted to get home as soon as possible.
The one I have chosen is without video; just a live sound recording of Leonard Bernstein and the Vienna Philharmonic, but it is a fine performance well worth listening to.

Here are excerpts from the 1973 recordings by Wolfgang Sawallisch and the Staatskapelle Dresden. If you want to download the complete symphony, or the entire set, you can do it very cheaply and legally from here

1st Movement (Introduction) - Ziemlich langsam   
1st Movement  - Lebhaft   
2nd  Movement (Romanze) - Ziemlich Langsam   
3rd  Movement (Scherzo) - Lebhaft   
Bridge 3rd to 4th  Movements - Langsam   
4th Movement - Lebhaft   

The Original (1841) Version
Very few recordings of the original version have been made, and I can't find anything worth recommending on the Internet.  There is a link to a Naxos recording which says it's the original version, but having listened to it I can state categorically that it isn't as Schumann wrote it.
I have a recording, no longer available,  made in 1988 by the RSO Berlin conducted by Georg Scmohe, which IS the original version, complete with guitar in the second movement. Which is better, the 1841 or revised versions is an interesting debating point as in my opinion both have their plus and minus points.  Here are a few extracts to help you make your own mind up.
 
1st Movement (Introduction) - Andante con Moto   
2nd Movement (Romanze) -  Extract 1   
2nd  Movement (Romanze) - Extract 2   
3rd  Movement (Scherzo) - Presto   
Bridge 3rd to 4th  Movements - Largo   
4th Movement - Allegro Vivace   

Symphony No. 1  Symphony No. 2  Symphony No. 3  Symphony No. 4