Learn English – Difference between “did save” and “saved”

Firstly, I think your second example should be have saved, rather than have save.

To save up pretty much always means deliberately putting money aside for something. There will be a purpose, and a defined saving period: you choose when to start saving, and you finish when you (hopefully) reach a certain amount, and/or reach a deadline. The 'up' implies working towards something, even if it is not specified in the sentence.

To save can imply that you are putting money aside, gathering/gaining money, or not spending money. There may be a specific purpose, target amount, deadline, but there may not. It can be deliberate or not.

A simple rule: (in my experience of British English), if there is no target/purpose, you should say 'save', not 'save up'. If there is a target/purpose, 'save up' is more natural (although you can say 'save').

Examples:

• You get a 33% discount at a shop. You look at the receipt to see how much money you have saved [=not had to pay].
• You put 10% of your wages into a separate account each month. You look at the balance to see how much money you have saved [=accrued, no obvious purpose].
• You put 10% of your wages into a separate account each month so you can buy a house one day. You look at the balance to see how much you have saved up [=accrued, specific purpose
• You stopped buying takeaway coffee two years ago. You do a quick calculation to see how much you have saved [=not spent]. It's a lot!
• You are going on holiday next year and it's going to be expensive, so you are saving up [=deliberately putting money aside] for it.
• Your friend asks if you want to go for dinner. You would love to, but you say no, because you're saving up [=deliberately putting money aside] for a new car.
• Your friend asks if you want to go for a drink. You would love to, but you say no, because you're trying to save money [=generally trying to spend less money].
• You are hoping to have a baby in a few years, so you are saving up [=deliberately putting money aside because you know you will need it, even if you don't know exactly how much it will cost, or what you will buy]

Deliberately and on purpose can be used interchangeably to mean "intentionally".

But deliberate (adj) and deliberately (adv) also mean "done with care, methodically."

He made the sandwich very deliberately, carefully covering the bread with a layer of peanut butter and then evenly applying the jelly.