It could be these two lines.
IPAddress ip(192, 168, 0, 1); // IP address, may need to change depending on network
EthernetServer server(8888); // create a server at port 80
A majority of routers use 192.168.0.1 for the router's IP. You might want to change that.
Most browsers expect the web server to be on port 80. Yours is on port 8888. You would need to use this:
http://192.168.0.1:8888
edit: "Connection refused" is normally a router response, not the Arduino. If the Arduino will not respond, and you get that message instead, then it is probable the Arduino is assigned the IP that the router is already using. No duplicates allowed.