When it comes to your private savings it is all about sparen. This can mean to spend less or to put something aside. Einsparen means in this context to cut expenses (e.g. of a government or city). There is a subtle hint on the consequences (close libraries, fire people …) in einsparen.
However, sparen and einsparen are often used synonymously when it comes to cuts in public spending:
Die Regierung muss 10 Milliarden Euro sparen.
Die Regierung muss 10 Milliarden Euro einsparen.
Etwas einzusparen can also be the result of other measures like an investment, e.g.:
Mit den Automaten können wir 200 Mitarbeiter einsparen.