Saving money implies that you have have money to put in a safe place, in case something bad happens. For example, if you got paid £1000 every month, you might save £100 of that and put it in to your spare bank account, just in case you lose your job.
Being thrifty means that you don't have much money left to spend, so when you do spend money, you spend it very wisely. For instead of going to your usual clothes shop to buy new clothes, you might go to the charity shop where they sell very cheap clothes. "Thrifty" isn't really a word we use in the UK, it is used almost exclusively in the US, and they also have "Thrift stores" which are basically charity shops I believe, that take donations from people and then sell them at a really low cost.