Bitcoin wallets and Lightning wallets both send and receive bitcoin but do so in different ways.
A Bitcoin wallet broadcasts transactions (with a fee) via the Bitcoin network to all participants, awaiting the transaction's acceptance through the consensus mechanism, which involves the inclusion of the transaction into a mined block. This process can take several minutes to even days but averages around 10 minutes.
On the other hand, a Lightning wallet uses the Lightning Network, a web of peer-to-peer payment channels, to exchange and transact bitcoin. This approach is significantly faster (near instant) and less costly, making Lightning increasingly popular for small payments.
However, there are some notable differences between the two types of wallets. A Lightning wallet must be online to send or receive payments, while a Bitcoin wallet can passively receive payments when the sender transfers bitcoin to an address the wallet controls. Lightning also presents more complexity in its setup, as it requires the commitment of a specific amount of bitcoin upfront to create channel liquidity, which can then be spent.
For these reasons, many Lightning wallets, especially those used by less sophisticated users, are managed by a custodian. This custodian, who is proficient and online 24/7, processes payments on behalf of the user. An example of this service is the Wallet of Satoshi app.
Users typically initiate incoming and outgoing payments by creating or paying a Lightning invoice through a mobile app. This Lightning invoice, similar to a Bitcoin address, is a unique piece of data usually displayed as a string or QR code. However, it operates exclusively within the Lightning Network, and unlike a Bitcoin address, it cannot be reused.
Recommended Lightning wallets: Phoenix, Muun, Blockstream Green (Lightning coming soon)
TL;DR Bitcoin and Lightning wallets send and receive bitcoin but do so using different networks and tradeoffs. Bitcoin wallets send transactions via the Bitcoin network, waiting for consensus acceptance, a process that can take minutes to days. Lightning wallets, however, use the faster and cheaper Lightning Network, making them preferred for small payments. A tradeoff is Lightning transactions require an online presence and upfront Bitcoin commitment for channel liquidity, making them more complex. Consequently, many less savvy users rely on custodians to manage their Lightning wallets.