Think about customer service as reactive. And think about customer experience as proactive.
Customer service could be when a client is upset, you handle it, you react, you solve.
Customer Experience is Proactive
You reach out to a customer before they are upset, and make their lives easier. Even if they call agitated, you can make that client feel more appreciated and comfortable by treating them like a person.
Go off the script, and instead of putting a client on hold, talk to them while you look up information. Ask them about their day, and thank them for being a loyal customer.
Keep open contact with a client, and treat everyone like a VIP.