In the FE model there is no between variation as it has been removed - dummied out - and you cannot include any between variables.

There are more conceptual and practical reasons for making your choice - see these two papers

https://www.researchgate.net/publication/233756428_Explaining_Fixed_Effects_Random_Effects_Modeling_of_Time-Series_Cross-Sectional_and_Panel_Data

https://www.researchgate.net/publication/299604336_Fixed_and_Random_effects_making_an_informed_choice

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

Article Fixed and Random effects models: making an informed choice

As a rule of thumb, the choice of FE or RE should not be determined by the size of standard errors. If your estimates are bias then statistical significance does not matter. The choice should be determined by whether or not you think there are unobserved, individual specific (and time-invariant), factors that impact your outcome and are correlated with the X variable(s) of interest. If there are, use FE, if not, use RE.

Unfortunately, you will rarely know for-sure. So the safe option is usually FE. If the coefficient estimates differ dramatically between FE to and RE, that is suggestive that RE is bias.

One way to think about RE is that it is a GLS transformation of the OLS estimates to correct for a specific form of serial correlation. Therefore, it requires the same assumptions with respect to the correlation between X and unobservables.

There is one important exception. When your X variable(s) are measured with error. In this case the FE estimator may exacerbate the error-in-variable bias. It is even possible that the FE estimates are more bias than OLS.

Jesse: Thank you for those reasonable guiding points! I agree with you on prejudgment as this presents its own new biases. Back to the rule of thumb, there is an argument, particularly in panel data estimation, that when N < T, then FE is prefered to RE. Conversely, when N > T, then go for RE. Does this even matter?

Most of the work I do is with a very large N relative to T, am not familiar with this argument (although I have hear it). The short answer is no, it does not matter. The choice between FE and RE should be guided by the underlying economic model (i.e. are there unobservable fixed-effects that may be problematic?). Arguments that have to do with the structure of your data are probably made based on the asymptotic properties of the estimators, but have nothing to do with whether or not RE will produce a bias estimate. It may be worth noting, the micro-econometrics literature almost exclusively deals with samples for which N>T, and you rarely see RE used.  

Confusing arguments, like the one your mention, are often made in literatures outside of economics (and even some stuff inside economics). The reason (I think) is that they often do not have an underlying model to guide their analysis, so they are guided solely by the statistical properties of the estimators.

To be safe (from bias) just use FE. As long as you have good data (lots of within variation and no errors in your Xs) you will not be "wrong". 

You may want to look at this interchange which argues  for random effects modelling as it can achieve ( in its within and between form) all that fixed effects can achieve ( it gets the same results) and it can do much more because of its flexibility. 

https://www.researchgate.net/post/Is_there_some_other_method_other_than_hausman_test_to_decide_which_method_either_fixed_or_random_should_be_adopted_in_panel_data_analysis

Speaking of FE and RE differences, is there a way to make the FE model estimate inefficiencies that are not constant over time for each unit in Stata? I would like to see the evolution of inefficiency/efficiency for each of the DMUs, but most Panel Data models only repeat one value per unit across the entire period being analysed. I managed to change this for an RE model (see code below), but I'm curious if it can also be done for an FE model.

xtreg lnVA $xvar_cd, re

matrix bre =e(b)

predict ue, ue

sort id

generate RE_hat = (ue - _b[_cons])

quietly summarize RE_hat

generate double u_star_re = r(max)-RE_hat

generate double TE_re = exp(-u_star_re)

Thanks for your help!