Some filters have distinctive sounds or colour the output in a specific way, but theoretically they're all doing the same thing. In most cases though the effect is the same. If you know you only want to apply a single LP filter to a synth, it can be nicer or easier to use 'just a single filter VST', or an in-built one on the synth, rather than loading a whole EQ and only using one node as a LP filter. There's generally no reason for being able to do things in multiple ways in a DAW environment, it's just nice to have options to choose from. It’s a way to clean up your mix and / or to make the most important elements (such as kick and bass in EDM) to really stand out from the rest of the material.
Zooming out (cutting) is generally okay for image quality). High pass filter (or low cut filter) should be one of the most basic and essential EQ tools in your mixing arsenal. Generally though cuts are recommended over boosts for reasons (think zooming on a JPEG - zoom in too far (boosting) gives you jaggies. (Video 11) A shelving filter is designed to boost or cut the signal at the target frequency and c that boost or cut into lower or higher frequencies. A filter can be only boosting the signal somewhere, it's still a filter. Any 'changing of the signal' whether it's a cut or a boost is still a filter. Think of an EQ as a string of variable filters (can be changed from lowpass to bandpass to highpass at will) that you can visually see and move around easily.
