The near-equivalence relations between FF and cropped camera are the following:
FocalLengthFF = FocalLengthcrop * CropFactor
FstopFF = Fstopcrop * CropFactor
ISOFF = ISOcrop * CropFactor2
where most people know the first equation, but many forget the second and last equations.
The first equation explains how focal length needs to be modified to take into account the crop factor to maintain the same field of view. This is what most people know.
Now, if you make the focal length smaller but maintain the same F-stop, it means the lens will collect less light, because the diameter of the aperture opening is:
ApertureOpening = FocalLength / Fstop
...and to keep the fraction the same, both the numerator and denominator need to be changed by multiplying by the crop factor. This ensures the aperture opening and thus the ability to collect light stays the same.
Now, exposure and ISO are defined in such a manner that exposure is:
Exposure = ISO * ExposureTime / Fstop2
Exposure time is obviously the same on FF and crop cameras if you want to take the equivalent picture. Now, as I explained that to maintain the light-collecting ability, you must multiply Fstop by CropFactor. To maintain the exposure, you must therefore multiple ISO by CropFactor2. Is this a problem? No, because FF sensors are physically larger in terms of area by a factor of, you guessed it, CropFactor2, so you can multiply ISO by CropFactor2 without having any adverse noise effects, assuming the pixel size becomes larger, i.e. megapixel count is the same.
So, let's check:
- Field of view: maintained
- Light-collecting ability: maintained
- Noise level: maintained
- Exposure: maintained
Now, there are two other factors which might affect your choice of equipment. They are the depth of field (DOF) and background blur.
As @xiota explained, DOF formula is:
DOF = 2 SubjDistance2 Fstop CoC / FocalLength2
Subject distance stays the same, Fstop is multiplied by CropFactor, CoC (circle of confusion) is multiplied also by crop factor because sensor dimensions are larger by a factor of CropFactor. The denominator is multiplied also by CropFactor2, so depth of field (DOF) stays the same.
However, there's also another aspect, the background blur. My understanding is that background blur is:
Blur = FocalLength SubjMagnification BgDistance / (Fstop (SubjDistance ± BgDistance))
If SubjMagnification is unitless, the numerator has units of length squared. The denominator has units of length. So, blur has units of length.
Let's check what happens for an FF camera. FocalLength is multiplied by crop factor, but also Fstop is multiplied by crop factor too. Subject magnification is apparently size of sensor divided by size of subject. Size of subject stays the same, but size of sensor is smaller or larger by a factor of CropFactor. So, on FF, SubjMagnification is multiplied by CropFactor. So, Blur is multiplied by CropFactor. Thus, the blur disc size becomes larger, but so does the sensor size, so the blur disc occupies the same percentage of the sensor!
So, let's check the background characteristics:
- Depth of field: maintained
- Background blur: maintained
So, yes, the photos would be identical if you use an equivalent lens. However, do note that you can probably find a 80mm f/1.2 lens very easily (well, ok, it might be 85mm but close enough), but finding a 50mm f/0.75 lens might be a bit challenging. So, if you want to have lots of background blur, shallow depth of field and low noise, there's some benefit in using full frame: you probably can't find the lens you want for a crop camera!
If we go smaller still, and consider mobile phone sensors (crop factor of 7-8), you would need 10-11mm lens with F-stop of approximately f/0.15 - f/0.17. I'm sure you won't find such a lens!
Let's do a quick check on the validity of the near-equivalence relations. The Canon 17-55mm f/2.8 IS USM zoom weighs 645 grams. On full frame, it would be 27-88mm f/4.5. You can find 24-70mm f/4 IS USM lens that weighs 600 grams, and 24-105mm f/4 IS USM lens that weighs 669 grams. Filter thread size is 77mm on all lenses. So I guess they must be nearly equivalent, having approximately the same amount of glass.
However, the 24-70mm f/2.8 non-IS USM weight is 953 grams, so it clearly has more glass in it.
Also, consider for example Coolpix P1000. It is advertised as having 125x zoom, 4.3 - 539 mm, equivalent to 24-3000 mm. The F-stop is f/2.8 - f/8, but there is no "equivalent to" spec for F-stop, which the manufacturer conveniently forgot. Have you seen a 3000mm f/8 lens? I haven't, but it would be honking huge, being at least 3000mm/8 = 375mm in diameter. The manufacturer should have remembered to say that f/2.8 - f/8 is f/15.6 - f/44.5 equivalent. This demonstrates that people typically forget the equivalence relation for the F-stop, remembering just the relation for the focal length.