The simple version is that both Union and Confederacy cared about British neutrality for the same reason - both sides of the same coin so to speak.
The Confederacy was an agrarian society. Their two major crops were cotton and tobacco, much of which they exported to England. Their manufacturing capabilities were minor and they relied on the income they got from their trade with England to purchase many manufactured goods, especially weapons.
The Union had considerable manufacturing capability of their own, and wanted England to stay neutral for exactly that reason, so that they (the English) would not respond militarily to the Union blockade of Confederate ports, which effectively cut off the supply of arms, equipment and cash the Confederacy needed to survive.
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Hope this helps!
- Lexi