The difference between spectrum and RTA modes is how the information is presented.
In spectrum mode the frequency content of the signal is split up into bins that are all the same width in Hz. For example, with a 64k FFT length and 48 kHz sample rate the bins are 0.732 Hz wide. The plot shows the energy in each of those bins.
In RTA mode the bin widths are an octave fraction, so their width in Hz varies with the frequency. For example, a 1 octave RTA plot has bins that are 70.7 Hz wide at 100 Hz and 707 Hz wide at 1 kHz. The plot shows the combined energy at each frequency within each bin. This is closer to how our ears perceive sound.
The different presentations mean signals with a spread of frequency content will look different on the plot. The best known examples are white noise and pink noise.
White noise has the same energy at each frequency. On a spectrum plot, which shows the energy at each frequency, the white noise plots as a horizontal line. On an RTA plot it appears as a line that rises with increasing frequency, as each RTA bin gets wider it covers more frequencies and so has more energy, the bin widths double with each doubling of frequency so the energy also doubles, which adds 3 dB on the logarithmic plots we use to show level. White noise sounds quite 'hissy', we perceive it as having more energy at higher frequencies.
Pink noise has energy that falls 3 dB with each doubling of frequency. On a spectrum plot it is a line that falls at that 3 dB per octave rate, on an RTA plot it is a horizontal line as the energy in the signal is falling at the same rate as the bins are widening. We perceive pink noise as having a uniform distribution of energy.
Single tones are a special case, they will appear at the same level on either style of plot as their energy is all at one frequency, so on a spectrum plot they show as a vertical line, on an RTA plot they show (typically) as a bar of the width of the bin width at their frequency, but the height of the bar is the same as the height of the line on the spectrum as all the energy is at that one frequency.