A small change assumption is an assumption as the name suggest, wherein you neglect the changes to the concentration of a species.
The common ion effect is the lack of further dissolution of a salt for example into a solution already containing a high concentration of ions that make up the salt. I.e try to dissolve table salt (NaCl) into a beaker with a high conc. of Na+
You apply the Small change assumption in cases of common ions being present in solution because like i said above, if you try to mix more salt into already salt water it ain't happening. So you assume only a small change in the conc. of the salt.
Also the small change assumption can be applied to highly insoluble chemicals (like salts or acids with a very small Keq). So yh people use the small change assumption to solve the common ion effect q, but it is also applicable in other situations where the equilibrium of a system lies considerably to the left.
Basically, common ion is the situation. Small change is the action you take when faced with the situation.
Idk how to explain it better lmao sorry.