Difference between an “Ordinary Least Square (OLS) model” and a “Panel Fixed- Effects (FE) model”

I recently ran into a paper that described the following:

To test the robustness of each specification, we used a difference-in-difference (DID) estimator to control for time invariant factors that jointly affected control and treated units. We estimated the DID with i) an Ordinary Least Square (OLS) model and with ii) a Panel Fixed-Effects (FE) model.

It is my understanding that an OLS model is (or at least it can be) based on a panel data, where the covariates are treated as fixed effects, right? So what is the difference between the two models tested in the paper?

It is economists' speech for saying that for $y_{it}$ being the observation for individual $i$ at time $t$ and $x_{it}$ being the regressor vector for individual $i$ at time $t$, they ran two models: \begin{align} y_{it} & = \beta_0 + x_{it}^T\beta + \varepsilon_{it}, \end{align} which they call OLS and for which $\beta_0$ is the overall intercept, and \begin{align} y_{it} & = x_{it}^T\beta + \alpha_i + \varepsilon_{it}, \end{align} which they call a Fixed Effects Model and for which $\alpha_i$ is the individual-specific intercept. The terminology is somewhat confusing, since you still estimate your Fixed Effects Model using the OLS estimator, but you will transform the data first using the within-transformation: \begin{align} y_{it} -\bar{y}_i & = (x_{it}^T - \bar{x}_i)\beta + (\alpha_i -\bar{\alpha}_i) + \varepsilon_{it}, \end{align} which you apply to take out the individual-specific term $\alpha_i$. Since you expect it to be fixed across time, $\bar{\alpha}_i = \alpha_i$ and so the term vanishes when you run the actual regression. If you want to estimate the $\alpha_i$ terms, you can do that. Economists call the resulting model the Random effects model. This model has the problem that you need to estimate an additional parameter per individual, and thus your standard errors get larger as you have more individuals, which is why Fixed Effects are preferred if $N$ is large relative to $T. The difference is really only the inclusion of an individual-specific intercept term ($= \alpha_i$) to replace the overall intercept ($=\beta_0$). Usually, you try to capture individual-specific heterogeneity with the$\alpha_i$, so it serves as a kind of 'catch all' term. (E.g., if$y_{it}$is the IQ and$x_{it}$are some environmental factors, you could interpret$\alpha_i\$ as a kind of catch-all term for an individual's genetic predisposition and any other unmeasured individual-specific variables that influence the IQ)