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Each of my vectors consists of beta estimates for two separate models of the same data and the same number of explanatory variables. The question is asking whether the difference between these two vectors is different. I found my estimates and found the difference between the vectors but I'm having trouble on how the test would work. Any help is appreciated.

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The question is asking whether the difference between these two vectors is different

Your question isn't clear. According to the tags, I can only infer that you want to see if the two vectors you've obtained arise from the same distribution.

So, if you want to test $H_0: \mathbf{\beta_1} = \mathbf{\beta_2}$, testing this hypothesis is equivalent to testing the following hypothesis $$ H_0: \mathbf{\beta_1 - \beta_2 = 0} $$

Since I don't know your model or how your data looks like, if you know the variance of the distribution, you could consider performing a $Z^2$ (when you know the true variance) or a $T^2$ (when you know the sample variance).

If the two models you've trained are completely different, for example, if you have 3 variables $x_1, x_2, x_3$ and you want to learn the regression coefficients to predict $y$, you could have trained two different models

Model 1$: y = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon$

Model 2$: y = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_1^2 + \beta_5 x_2^2 + \beta_6 x_3^2 + \epsilon$

In such cases it makes no sense to compare the two regression coefficients vectors you obtain.

I also implore you to verify if the question enquired if the results obtained from the two models differ significantly in performance instead of comparing the coefficients themselves

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