For a straight conducting wire, the magnetic field curls around it in concentric circular field lines. In the case of a current carrying plate (of infinite spread), you can consider it as a tightly packed combination of an infinite number of parallel wires all carrying the same current, with negligible cross-section along a particular dimension so that the resulting arrangement is two-dimensional. In such a case also, the flow of current have the same direction as were in the wire. But the difference is that the flow of current is now through a surface.
To obtain an expression for the magnetic field due to an infinite conducting sheet, we assume that the current through each wire is $I$ and there are $n$ wires per unit length.
We have the Ampere's law:
$$\oint \textbf{B}.d\textbf{l}=\mu_0 I_{enc}$$
The line integral is through a rectangular loop surrounding the current sheet whose plane is perpendicular to the plane of sheet. $I_{enc}$ is the current enclosed by the Amperian loop.
As you can imagine the magnetic field lines lie in a plane perpendicular to the direction of current through the wire. So here in our sheet,in the two sides perpendicular to the sheet the direction of magnetic field is perpendicular to the $I_{enc}$
Hence
$$\textbf{B}.d\textbf{l}=0$$
and the only contribution is from the two parallel sides and so we write
$$2\int Bdl=\mu_0 I_{enc}$$
or $$2BL=\mu_0 I_{enc}$$
or $$B=\frac{\mu_0 I_{enc}}{2L}$$ where $I_{enc}=INL$
Hence $$B=\frac{\mu_0 IN}{2}$$
which means the field strength now is not dependent on the distance from the current carrying sheet. The same analogy you can find in the case of electric field due to a charged infinitely long sheet.
In the case of magnetic field due to a conducting wire, the field decreases as a function of distance from the wire. When such wires are combined to form a sheet, the magnetic field is independent of the distance from the sheet. You can imagine the magnetic field extending out to infinity (of course t is forming a closed loop of infinite extent). This is how the geometry varies.
Since the field has changed there will be a slight change due to the difference in field value for the Lorentz force also.