"When in reality, the MLE is far away from the parameter value assuming the null hypothesis"—if distance of the M.L.E. from the null is your measure of discrepancy with the null then you can use it as your test statistic. The point of the score test is to maximize power against alternatives close to the null (when the M.L.E oughtn't to be too far away).

Consider, say, a single observation $x$ from a Cauchy distribution with unknown location parameter. With a test of $H_0: \mu = 0$ vs $H_1: \mu = k, k>0$, any observation $x\gg k$ is only marginally more probable under the null than the alternative (& $x\ll k$ only marginally less probable): there's no uniformly most powerful test of $H_0: \mu = 0$ vs $H_1: \mu > 0$, & a locally most-powerful test might be what's wanted—trading off power against distant alternatives for power against near ones.

In other cases, say i.i.d. observations following a Poisson distribution, the score & likelihood ratio will order the sufficient statistic identically (for one sided tests at any rate), so the *exact* tests will be equivalent. If you're relying on an asymptotic approximation to the distribution then you'll get somewhat different results with the two as [kjetil's answer](https://stats.stackexchange.com/a/635155/17230) points out.