I was told that they are not the same but without any explanation. What is the difference between the two, if any?
I have no reference on this topic other than all the weird people I've worked with / for over the years.
Solve for the user, validate that you've addressed the business need.
Solve for the business need, validate that users respond well.
Too subtle to be practical
Linguistically, the difference is too subtle to use reliably. The only way to know with any certainty what someone means is by analyzing their design process. Where do they start the discovery? Where are they centered?
For the record, I think both are incomplete concepts and we should hear a lot more about activity-centered design.
At least in the UX field, we use "user-centered design" when the user goals, tasks, environments, feelings, etc are taken as a key factor in the design process. There isn't at least an already established term as "user-centric". So, it not been an already "known term" by the field, it would be reasonable just to take it as a synonym.
If somebody on this field tries incorporate to use these two different terms, they may be going against the basics of this discipline by dismissing the importance of key UX features as: familiarity, convention, labeling, avoiding ambiguity, "don't make me think", etc.
If in your particular environment everyone use them as different terms (or if you have a "not-open-I-know-it-all-superior") you don't have too much choice but to learn how they use it. Otherwise I'd try to suggest another term to avoid future confusion.
I think these two terms reflect a sliding scale of 'user-centricity' when it comes to design. Just as the Agile Manifesto is more of a philosophy and approach rather than a prescribed set of methodologies and tools (which we often forget), UX is really a particular perspective or approach in design.
So rather than trying to split hairs defining what shade of grey each term refers to, it is better to try and find out how much of the requirements, design decisions, development decisions and testing process/outcome is going to be directly relating or coming from the user. Then you can decide what they mean by the terms that they use.