For reference, I'm using a Nikon T-2000U microscopy with a Hamamatsu Orca-Flash 4.0 CMOS camera. I'm capturing brightfield and epifluorescence images of cancer cells.
One likely difference is that you might have different artifacts associated with the objectives. For example you might have some abberation especially around the edges of the field of view, or the illumination might be uneven. The intermediate magnification lens would take only the center of the view through the objective and magnify it, avoiding abberations from the edge of your 60x objective and it's possible this looks better than the 90x objective. But, the intermediate 1.5x magnifier might be introducing its own artifacts. Or, maybe one objective isn't as clear as the other one and you would be better off with the one that gives better images. Especially chromatic abberations can vary between different objectives.
Depending on your application it may or may not make a difference, e.g. if you are doing some sensitive quantification that relies on an even illumination. You might just have to try both, and see which one gives you better images.
Check out Olympus or Nikon's microscopy resources (https://www.microscopyu.com/articles/optics/objectiveintro.html) as they have some good discussion and more technical details as well.
The importance here is the numerical aperture (NA) of the lens. It is what determine the resolution of your image. If you are using a high NA 60x objective, you might not notice a difference between a 60x with a 1.5x magnifier and a 90x objective.
If your 60 objective has a low NA, the resolution will be low so increasing the magnification with the 1.5x lens will just make a larger / blurrier picture. If your NA is higher, then you will probably not notice a difference between a 90x lens and a 60x lens with a 1.5x modifier in the path length.
The easiest way to make the practical difference between collecting images using a 60x objective with 1.5x intermediate magnification versus using a 90x objective directly visible and measurable is the use of an original micrometer (1 mm). This preparation shows the real difference in length and width, and potential distortions. The choice of camera is absolutely independent.
Please find enclosed the direct comparison of two images taken with my original micrometer (one millimeter divided into 100 divisions) with a 50x magnification and a 100x magnification objective.
Please find enclosed the direct comparison of two images taken with my original micrometer (one millimeter divided into 100 divisions) with a 50x magnification and a 100x magnification objective.