Pros of Non Exclusive Publishing
Larger potential audience reach. With your music in multiple libraries you are increasing the exposure of your music.
Commercial Freedom. You are still in control of the rights to your music so you can do with it what you wish. Want to sell on iTunes or bandcamp? No problem. Want to include an existing piece of yours into a film score you are writing? No problem.
Multiple revenue sources. It can be beneficial to have your music split across multiple libraries so that you don't fall into the trap of having all your 'eggs in one basket'. If you are just with one exclusive, this could lead to financial insecurity in the future.
Cons of Non Exclusive Publishing
More Work - It can take a lot of work to upload each track to various sites. Every publisher has different upload requirements with some requiring you to add your own audio watermarks etc. For each library you add a track to, you are multiplying the administrative aspect.
Smaller Sales Percentage - Often sites such as Audio Jungle will offer a smaller percentage of the sale to the composer.
Broadcaster Administrative issues. Various digital technologies have recently been developed which can identify a track in order to benefit royalty collection from Broadcasters. A non exclusive track in multiple libraries could potentially cause legal complexities among broadcasters/publishers in the future. For this reason, some broadcasters/production companies are choosing to avoid music from non-exclusive publishers and instead focus on exclusive.