Learn English – Copy vs photocopy difference

Indeed, to wander and to roam are synonyms, and they both can mean to move about aimlessly or without any destination, often in search of food or employment.

However, to wander has some additional meanings that to roam doesn't:

• Lose clarity or turn aside from the main subject. (Her mind wanders.)
• To be sexually unfaithful.
• To go an indirect route or at no set pace. (We wandered into town.)
• To move in a sinuous, spiral or circular course. (Synonym with meander.)

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]