Lightweight bitcoin clients differ mainly in that the blockchain is not stored on your computer, so they rely on a server to receive and send transactions.
Advantages are:
- Allows almost the same level of security as locally stored desktop wallets
- Work well in environments with a slow or limited internet connection, such as mobile phones and rural areas with dialup.
Disadvantages are:
- Have to trust the server to be online reliably
- Have the privacy disadvantages of browser and online wallets
- No options available for iPhones, even rooted ones through Cydia
The main options for this type are Electrum for the desktop and Bitcoinspinner for mobile phones (Android only).