Individually, you don't need to store the whole Blockchain. If you have no need for transaction information for addresses not associated with your wallet file, you don't have to use it. You can easily just prune the Blockchain. The disadvantage is that you can't access any transactions related to any other address that is not associated with your wallet.
You need to download the whole Blockchain though. It is unsafe to download only parts of it and expect it to be valid. At all times, a portion of the nodes must have the full Blockchain to facilitate their peers in synchronizing.
Pruned mode is paradoxical because you need to be able to download the entire thing at least once... which kind of defeats the purpose. At best you would be able to save space afterwards. If we could engineer something that is able to download only the last GB which would be a reasonable amount since trying to go "1GB back in time" would require an huge amount of money for such a hashrate attack... but I don't see how that's even possible. You would need some epic wizardry engineering to pull it off without opening potential exploits.
You can use wallet which not store all blockchain on your computer, look at Electrum
SPV wallets aren't safe... and I don't think we will ever be able to get past having to download the entire blockchain and validate it at least once.