There is not much practical difference. There are many cases where the u[x][y][t] forms are particularly convenient, but not necessarily for the derivative application you show.
For derivatives, I prefer the operator-like forms:
Derivative[0, 1][ArcTan][x, y]
Derivative[0, 1][ArcTan]
Derivative[0, 1][ArcTan][5, y]
Derivative[0, 1][ArcTan][5, 12]
When I want to pass a function as an argument, the u[x][y][t] forms are particularly convenient, consider this little tool to plot tangent lines:
tangentLine[f_][x_][t_] := Module[{z, derivative},
derivative = D[f[z], z] /. z -> x;
Arrow[{{x, f[x]}, {x + t, f[x] + t derivative}}]
]
or
tangentLine[f_][x_][t_] :=
Module[{derivative}, derivative = Derivative[1][f][x];
Arrow[{{x, f[x]}, {x + t, f[x] + t derivative}}]]
Manipulate[
Plot[f[z], {z, 0, 4},
Epilog -> {PointSize[0.05], tangentLine[f][x][t]}],
{x, 0, 4},
{t, .3, 4},
{f, {Sin, Cos, Function[{u}, u/(1 + u)]}}
]
Using the f[x][y][t] makes the code more tidy, I think.
