You will get the APS-C fans claiming their cameras are as good as FF cameras.
You will get the FF purists who claim the APS-C cameras are toys.
And then you'll always have the M4/3's fans who claim their tiny sensors are better than either of the bigger brothers.
Truth is, FF is definitely the king of resolution, but many do not NEED that much resolution so smaller sensors will fill their needs just fine. And at a lower cost for the camera and crop sensor sized lenses.
The only thing that matter is what does the job for YOU!

Digital camera sensors convert light to electronic signals. The signals are then processed, amplified and converted to binary data. The image is constructed from said data. If two cameras have the same pixel count but one is full frame and one is crop frame, here's the difference. The pixels, the photo sensitive diodes that gather the light, on the full frame are larger than on the crop frame sensor. The larger the pixel, the more light it can gather and convert into a electrical signal. The stronger the signal, the less amplification needed when converting the signal into binary data. Lowering the amount of signal amplification needed, lowers the amount of electronic noise associated with the amplification, thus less noise converted into the binary data.
Another advantage of the larger sensor is the spacing between each pixel. The closer together the more susceptible the pixels are to heat generated noise. Since the full frame pixels are farther apart, less heat related noise.

The newer generation of digital cameras possess very powerful processors. These high speed processors are capable of detecting and differentiating between actual image data and a good bit of the noise data and filtering it out. That's why you can get pretty darn good images at ISO 16000 when in the past the usable ceiling was ISO 1600.

Better diodes in the pixels and better processors converting the signals to data; none of it amounts to a hill of beans if the glass put in front of the sensor is incapable of getting the details in the light to the sensor.

Today's crop cameras with their high end sensors and super fast processors are capable of producing images almost as good as a full frame but they still can't match the larger pixels light gathering ability, that's just simple physics.

Here are a couple of interesting articles by Roger N. Clark where he compares the low-light performance of a FF DLSR and an APS-C camera. Specifically he compares the 5D Mark III to the 7D Mark II. Both have similar size pixel counts. Essentially, he says the state-of-the-art is such that he would select the APS-C camera (7D mark II) over the full-frame camera. This is contrary to the conventional wisdom you hear on this web-site, but I think you will find he analysis credible. www. clarkvision.com/articles/does.pixel.size.matter/#Conclusions and www.clarkvision.com/reviews/evaluation-canon-7dii/index.html.