What the experts say
"The interviews gave us our starting point for the survey questions by helping us identify the kind of feedback we needed to gather from our customers. Our second round of customer interviews took place after we had gathered all our survey responses. These interviews helped us take the survey data and turn it into a story, a qualitative narrative we could use to provide context to the survey's quantitative data."
— An extract of a case study from Intuit Mailchimp in the book Surveys That Work by Caroline Jarrett
My two cents
Start with interviews, confirm with a survey
A series of interviews is a great starting point to understand a topic and discover the big problems or opportunities. But interviews then don't help answer the question: is this a problem most of our customers have? That's when a survey is perfect.
A good survey will then help you reveal which of the problems you found are the most important, the most present in the majority of the people you serve.
Combine both methods to talk to the two types of decision-makers
Having both will give you:
The numbers that make it possible for people who have an excel like brain to trust your research.
The stories and emotions that make it possible for people who are more on the creative and intuitive side of things to trust your research.