The Fourier transform of a continuous sine wave is a single Dirac delta function located at the frequency of the sine wave and reflected around zero frequency into the positive and negative frequency spaces.
ListLinePlot[Abs[Fourier[Table[Sin[2 \[Pi] 1 t] , {t, 0, 5, 0.001}]]],
PlotRange -> {Automatic, {0, 40}}]
Note the symmetric spikes around list element 2500 in the above plot of a sine wave with frequency of unity.
If you reduce the resolution of the time steps:
ListLinePlot[Abs[Fourier[Table[Sin[2 \[Pi] 1 t] , {t, 0, 5, 0.1}]]],
PlotRange -> {Automatic, {0, 40}}]
the Dirac delta appears smeared out across a range of frequencies, this is an effect of the discrete nature of this transform.
I suspect there is an issue in the continuous case when using FourierTransform in that DiracDelta does not resolve to a numeric value when plotting, so you don't see the spike in the continuous form of the plot.
The result you obtain with when using Sin[x] UnitStep[x] in the discrete case is equivalent to Sin[x] as UnitStep[n] evaluates to 1, so use of the UnitStep results in no modification to the Sin function.
In the continuous case, Sin[x] UnitStep[x] does not evaluate to Sin[x] but a truncated sine wave. Sharp discontinuities, such as those introduced by unit steps, cause a smearing in the frequency domain. I suspect this is the origin of your broad spectrum like plot for the continuous case as can be seen by examining the Fourier transforms of the two expressions.
FourierTransform[Sin[t], t, \[Omega]]
$$i \sqrt{\frac{\pi }{2}} \text{DiracDelta}[-1+\omega ]-i \sqrt{\frac{\pi }{2}} \text{DiracDelta}[1+\omega ]$$
FourierTransform[Sin[t] UnitStep[t], t, \[Omega]]
$$-\frac{1}{2 \sqrt{2 \pi } (-1+\omega )}+\frac{1}{2 \sqrt{2 \pi } (1+\omega )}+\frac{1}{2} i \sqrt{\frac{\pi }{2}} \text{DiracDelta}[-1+\omega ]-\frac{1}{2} i \sqrt{\frac{\pi }{2}} \text{DiracDelta}[1+\omega ]$$
One option might be to replace DiracDelta with its discrete counterpart DiscreteDelta which evaluates to 1 at its location.
Table[DiscreteDelta[n], {n, -2, 2}]
{0, 0, 1, 0, 0}
FourierTransform[Sin[t], t, \[Omega]] /. DiracDelta -> DiscreteDelta
$$i \sqrt{\frac{\pi }{2}} \text{DiscreteDelta}[-1+\omega ]-i \sqrt{\frac{\pi }{2}} \text{DiscreteDelta}[1+\omega ]$$