Overall classification of frequencies in the radio spectrum
Classification of waves according to their frequency is just to determine groups with similar properties. However there is no dramatic change when leaving a class and entering the next one, this is a continuous and smooth change. So classifications are useful, but somehow limited in their use.
The classification you mention uses band limits in "3" (30, 300, 3000...) MHz, which corresponds to limits in wavelength of 10 m, 1 m, 10 cm, etc, as frequency and wavelength are linked by the formula: $\lambda = c / f$, where $\lambda$ is the wavelength in m, $c$ is the light speed (about 300,000,000 m/s) and $f$ is the frequency in Hz.
There are other classifications, e.g.
Names given to allocated segments of the radio spectrum
In these bands, sub-bands are allocated to different users. Allocations are agreed under the auspices of the International Telecommunication Union (IUT).
Aviation has got sub-bands in "VHF" and "UHF", and other users too. Sometimes sub-bands are shared between different users. Each industry has its lingo:
We know very well the "FM band", another fancy name for a frequency interval in VHF, just before the aviation VHF sub-band, up to 108 MHz. This interval is used for broadcasting radio, where the frequency is modulated by the audio signal. Also known as "CCIR band". More here.
Radioamateurs often refer to VHF as "144 MHz", and UHF as "432 MHz", because their allocated portions in these bands begin at these frequencies. 144-146/148 MHz will also be understood as the "2 m band". More here.
In aviation, it's usual to talk about "VHF" and "UHF" to refer to those segments that are "mostly" in the VHF band or in the UHF band.
VHF/UHF in aviation
To answer your question of frequency differences:
First there are actual VHF and UHF bands as defined by physicists, with limits in "3".
Then there are the slots allocated to aviation in the actual bands, they have no particular names, but users of these slots give them names, that are sometimes confusing. "Aviation bands" include:
- 108 - 112 MHz NAVAID (VOR, ILS Localizer)
- 112 - 118 MHz NAVAID (VOR, LASS and SCAT-I))
- 118 - 137 MHz VHF Air/Ground Communications
- 62 - 174 MHz Fixed, mobile Communications
- 225.0 - 328.6 MHz UHF Air/Ground Communications
- 328.6 - 335.4 MHz NAVAID (ILS Glide Slope)
- 335.4 - 399.9 MHz UHF Air/Ground Communications
- 406.0 - 406.1 MHz Satellite Emergency Position Indicating Radio Bcn
Source: FAA order 6050.32 (page 29).
FAA has chosen to see the 225.0 - 328.6 MHz segment as being UHF, even if it starts before 300 MHz. This has no technical impact or implication.
Frequency allocation in the US
In the US, the radio spectrum is allocated to users and final use according to this map (year 2016):
Frequency allocation in Europe
In Europe, spectrum allocation is under ECC responsibility. Current status can be found in "ERC Report 25", aka the European Common Allocation (ECA) Table, which has also a search frontend by country.