How do 'econometric' explanations differ from 'economic' ones?

Econometric and economic explanations are not some standardized terminology so it is often best to clarify with the person asking you but generally speaking, when someone asks for economic explanation they are looking for explanation in terms of an economic theory or underlying economic relationships. If someone asks for econometric explanation they are likely looking for an statistical explanation.

An example related to changing coefficients lets suppose we have models:

$$ Wage = 0.5 + 5 EDU $$ $$ Wage = 0.6 + 10 EDU + 6 EXP $$

Where EDU is education and EXP experience and lets also say we know $cor(EDU,EXP) neq 0$.

An economic explanation would be that from economic theory we know that wages depend on your marginal productivity which is some function of both education and experience. In the first regression we include only education and do not control for what the peoples actual experience is. Since people who have higher education might have less experience on average than people with less education (due to all that time spent in school) so we expect the coefficients to change when we actually control for the fact that not all people have the same experience.

An econometric explanation would be simply that we know from econometrics that in the more parsimonious model (assuming the more general model is true) the estimated beta coefficient in this case will be given by

$$E[hat{beta_1}]= beta_1 + beta_2 frac{sum (EDU_{i}-bar{EDU})(EXP_{i}-bar{EXP})}{sum (EDU_{i}-bar{EDU_1})^2}$$

we know that $cor(EDU, EXP) neq 0$ and that $beta_2 neq 0$ so we also know that coefficient $beta_1$ must change between the regressions.

However, again note that your question is a soft-question depending on where you study teachers might have slightly different understanding of this so you should always ask. Furthermore, you should also inquire on how rigorous you have to be when providing both economic and econometric explanations. Depending on the level of your class and university overall simple textual economic explanation like I provided here would not suffice and the econometric one could be arguably done better. The above is just quick example in order to make it easier to understand differences between them.