They are not equivalent.
In the linear regression model (matrix notation),
$$ y = X\beta + u,$$
the OLS estimator for $\beta$ is
$$\hat \beta_{OLS} = \beta + (X'X)^{-1}X'u.$$
Then
$$\mathbb E\left(\hat \beta_{OLS} \mid X\right) = \beta + (X'X)^{-1}X'\mathbb E\left(u \mid X\right).$$
Under the assumption that is usually called "mean-independence of the error term from the regressors", or "strict exogeneity of the regressors",
$$E\left(u \mid X\right)=0,$$
we get
$$\mathbb E\left(\hat \beta_{OLS} \mid X\right) = \beta$$
And as another answer pointed out, we then also get by the law of iterated expectations,
$$\mathbb E\left(\hat \beta_{OLS} \right)= \mathbb E \Big[\mathbb E\left(\hat \beta_{OLS} \mid X\right)\Big] = \mathbb E(\beta) = \beta.$$
The reason that econometrics uses conditional arguments is that with observational data, the regressors are stochastic. But the original framework in which linear regression and least-squares were developed, was controlled experiments, where the regressors were "deterministic", namely, their values were decided upon by the researcher (hence the name "design matrix" that is still used sometimes). There, there was no need to use conditional arguments, because, being deterministic, the regressors were not/could not be characterized by a disgtribution, to have moments, etc.
When we condition on a random variable, we treat it "as though" it is "fixed/deterministic".