They seem to represent the same idea as far as I can tell - can they be used interchangeably?
In discipline of Statistics, the terms "Independent Variables" and "Degrees of Freedom" are two distinct concepts.
Independent variables generally refers to a set of variables (traits) that are independent from one another. For example, if we take a random sample of people and record their age, we do not expect that the age of one individual influences the age of another individual.
Per wikapedia degrees of freedom refers to "the number of values in the final calculation of a statistic that are free to vary." For example, in a situation where we are trying to calculate the variation of the age of our sample, this would be $(n-1)$.
This may not be the case in another discipline, but you would need to ask someone of that field to know for sure.