Why do some wallets not need syncing? I recently downloaded the Bitcoin Core wallet, and was mildly inconvenienced by the fact that I had to spend 4 days downloading 200 GB, but I just beared with it. 

However, I soon discovered that some wallets like Electrum and Bitcoin Wallet don't need syncing at all, which makes me curious about why some wallets need syncing and others don't, and what are the benefits to syncing as it takes up a lot of space, time, and bandwidth?
In brief, so-called lightweight wallets have to trust the nodes they communicate with, full-node wallets don't have to trust other nodes.

Bitcoin's creator, Satoshi Nakamoto, wrote about Simplified Payment Verification (SPV):


  As such, the verification is reliable as long as honest nodes control the network, but is more
  vulnerable if the network is overpowered by an attacker. While network nodes can verify
  transactions for themselves, the simplified method can be fooled by an attacker's fabricated
  transactions for as long as the attacker can continue to overpower the network.