Fixed effects, first differences interpretation
The biggest difference between (1) and (3) is that (3) has incidental linear trends while (1) has common time effects. Differencing (1) gives

(1a) $\Delta \log(uclms_{it}) = (\theta_t - \theta_{t-1}) + \beta_1 \Delta ez_{it} + \Delta \epsilon_{it}$. 

Comparing this to (3) you see that (3) has $\gamma_i$ in place of $\theta_t - \theta_{t-1}$ in (1a). So three things are different: (A) As you said, it is FE vs FD; (B) the trend is free (using dummies) in (1), while linear in (3); (C) the trend is common in (1) but is individual-specific in (3).

It would be interesting how much difference is introduced by each of these three points.

About why FD is useful, well it is simple and informs us about functional relationship among variables in differences. So I'd say why not.