I'm checking on both Coinbase and coinmarketcap.com and according to Coinbase BTC's value is €4,784.84. According to coinmarketcap.com BTC's value is $5630.32
Why is this?
I think you just explained it in your post yourself. The coinbase value that you're looking at is a Euro evaluation. The coinmarketcap value that you saw was a USD evaluation. You should try to change some settings to set the base currency for these conversions equal to the currency used in your location to avoid further confusion. Also, it's a good idea to use Preev to track bitcoin prices over coinbase, for example.