Is the difference between full- and crop-frame DSLR footage noticeable and disrupting?
There are several sources of difference, but you should be able to handle them all with some work.  The first difference will be depth of field.  A full frame camera has a shallower depth of field at a given angle of view because it can achieve the same angle of view with a longer lens.  You can compensate for this by using a smaller aperture (higher f/value) for the full frame camera to make the full frame field of view wider, or you can simply ignore it as differences in depth of field between shots won't be that noticeable.

The next major issue will be color reproduction.  Different camera models will likely have a different feel in terms of their white balance.  If they are both balanced to the same grey target, it probably won't be that far off, but there will be some differences in shadow detail and the like, so you may need to do color grading of the footage to match the color of the footage.  This is typical anyway as color grading should be done to achieve the final look you want for the color.

Finally, you will need to be mindful of the crop factor of the APS-c camera which will result in a smaller field of view for similar lenses.  You will also want to consider the relative quality level of the lenses being used on each camera.  These may have a noticeable impact on the overall quality of the final video as well.

So, overall, there shouldn't be any issues that can't be adjusted for, but be aware of what you are doing and be prepared to make some basic corrections in post to adjust the footage to look similar.  You also want to make sure you use similar export frame rates and compression options so as to ensure similar file quality.  It is worth noting that most of these distinctions apply for any two different cameras though, not specifically crop vs ff.