so a formula is like a dead equation?

I'd say an equation is anything with an equals sign in it; a formula is an equation of the form A= stuffA= stuffA={\rm\ stuff} where AAA does not appear among the stuff on the right side.

I think that over time the distinction is lost. My math teacher, 35 years ago stated "formulas are used in chemistry, in math we have equations". To this day, the word 'formula' in math seems wrong, but I'd accept it's used commonly.

An equation is meant to be solved, that is, there are some unknowns. A formula is meant to be evaluated, that is, you replace all variables in it with values and get the value of the formula.

The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.