Categorical variables

I have a simple question that I cannot figure out. For a multilevel categorical independent variables (let's say four levels), what is the difference in modelling the variable as 3 dummy variables in comparison with coding it as four levels within one variable in a regression model? Are they equivalent and is there an advantage or disadvantage to either method (is the regression equation the same?). Thanks for your responses.


Less is more. Stay pure. Stay poor.
Curious how many obs are within each group? MLM control for both within and between group variance. Not controlling for it can increase type I errors. MLM also allow you to model random intercepts for groups and add random effect for groups.
I mean excluding random vs fixed effects, I was wondering whether there is a difference between using three dummy variables when you input into statistical software versus if you specify four groups and let's say set one of the groups as the reference?