Viruses on their outside are a collection of well fitting proteins. It turns out the shape of a protein is very important, it lets the protein do its "job". Or as much of a job as a physical shape can have. Think of a hammer, its shape makes it very good and driving nails but not very good at smoothing concrete. These viral protein shapes allow the virus to attach to human cells, open the cell wall and "inject" the malicious genetic code to the cell. Antibodies attach to those protein shapes and can rip apart the virus, make it easy for immune cells to find and destroy, and/or prevent the virus from attaching to human cells.

Each mutation in a virus alters the proteins and their shapes a little. Too much mutation and none of the parts fit and it is no longer self replicating. So asking how different does it need to be is a very difficult question to answer. A little bit and the various shaped antibodies the body produces will still bind to some of the virus's protein shapes, a little more and it might stop attaching to human cells (though it might attach to a different animal cell and thus you've found a variant that is ready to hop species), and too much more and now you've either killed the virus or it's something new entirely.

Tldr: these mutations affect the very being of the virus, too many and it stops being the virus it is.