Assuming the spectrum is sufficiently interpolated to see the detail within the resolution bandwidth (which is a reasonable assumption given the question), and the waveform isn't further windowed beyond the rectangular window (as I would argue that any such windowing that is applied is equivalent to the presence of multiple tones closely spaced given windowing is an applied amplitude modulation) then one obvious way to quickly distinguish these two cases is by observing the width of the main lobe (as well as all other features such as null locations, or non-existence of the nulls, but the main lobe width would be robust under noise conditions)-- given the kernel of the rectangular window (the DFT of the window ) convolves in frequency with each tone. If there is one tone we would see the kernel alone. If there are two tones spaced closer to together, then both tones would convolve with the kernel, and therefore widen the result. (The kernel for a rectangular window in time is the Dirichlet Kernel in frequency, basically an aliased Sinc function, so if the spectral peaks are the same width as the Dirichlet Kernel, which is known, not computed, then we can easily declare that peak to be a single tone).
If sufficiently interpolated and noise free then we can also quickly detect this by the lack of frequency nulls (although the nulls are soon buried by any noise so this would not be a robust test). If the spectrum wasn't interpolated and the frequencies are so close that a bin width increase is not obvious, then we could compare the phase and magnitudes of the leakage values in all other bins to what would be expected with the underlying Dirichlet Kernel, under noise conditions we could use all bins and a correlation to what would be expected for a single tone to provide a confidence of one tone or not.
Further Details on using Main Lobe Width Measurement
This process of main lobe width measurement would be precise in confirming a single tone down to very small frequency offsets in the case of a single exponential tones of the form $e^{j\omega t}$, but will have a lower offset limit for the case of real tones due to the interaction of the two exponential tones that each real tone contains ($cos(\omega t) = 0.5(e^{j\omega t} + e^{-j\omega t}))$. In which case we would see even for a single tone a small variation in main lobe width versus frequency that becomes more pronounced as we approach the Nyquist boundaries. The effects of this, and interaction from any other tones further away for the same reason, become significantly reduced if the waveform was windowed prior to computing the DFT (which can't be assumed based on what the OP has stated, and as I already mentioned I would argue the result of windowing a single tone results in the presence of multiple closely spaced tones!). To show the practical extent of using the main lobe width approach, I plot the variability of the main lobe width versus frequency for the rectangular window case at the bottom of the post, which provides insight into the practical limitation of how close we can distinguish multiple tones (pretty close! down to 0.02 bins for most of the frequency range, and 0.2 bins for all frequencies except the end points).
To demonstrate this effect of simply measuring the width of the main lobe for the case of an interpolated spectrum, and its relative insensitivity to noise to address hotpaws comment, see this example below with a rectangular window and two equally leveled tones that are within one bin width (the resolution bandwidth of the rectangular window is 1 bin, so more than that would not be sufficient for "close"):
Here is the comparison of the composite signal with and without noise, where the noise is only 6 dB lower:
(And in this case, even if the spectrum was not interpolated, meaning the only samples available are at the integer bin locations; we can see how the increased width of the main lobe would still be obvious through inspection)
For further details on the limitations of the main-lobe width measurement for a real tone, the following plot below shows the variation of the main-lobe bin width at -3 dB for a rectangular windowed 128 point DFT (interpolated by zero-padding out to 12800 points), where the "truth" given by a single exponential tone approaches 0.886 bins for large $N$ (where $N$ is the number of samples, derived from $2\omega/\pi$ for $Sinc(\omega) = \sin(\omega)/\omega = 0.707$, and can be exactly determined for any size N from the Dirichlet Kernel, $D(\omega)$, from $2\omega/\pi$ for $D(\omega) = 0.707$, where:
$$D(\omega) = \frac{\sin(N\omega/2)}{N\sin(\omega/2)}$$
and $\omega$ is the normalized radian frequency in units of radians/sample : $\omega \in [0, 2\pi)$ for $N \in [0, N)$
With the results shown in the plot immediately below:
As given by the convolution, this bin width will be increased by the frequency separation of two tones that are less than 1 bin apart, and as long as that separation is greater than twice the variability of a single tone, it would be detectable (for example the variability when we are more than 10 bins away from the Nyquist boundaries is less than 0.01 bins, which holds as N is increased). This plot below was simulated but could be analytically predicted by the interference pattern of the underlying Sinc function that becomes the Dirichlet Kernel through aliasing. This demonstrates the practicality of using this for distinguishing a single tone from multiple tones closely spaced.
