As far as I understand, Lightweight node are like SPV wallets, which include all mobile wallets.
They are very important for adoption.
If you want to buy a coffee, you don't to run a full node on your phone.
Just use some wallet like coinomi/samourai/mycellium etc, scan the QRCode and it´s done. Even 1 confirmation is OK.
Running a full node is important for people who own business and want to validade all transactions by themselves, without relying on third party to know if he has received is money or not.
These kind of services were even described in bitcoin whitepaper, at Chapter 8 - Simplified Payment Verification (SPV)
8. Simplified Payment Verification
It is possible to verify payments without running a full network node. A user only needs to keep
a copy of the block headers of the longest proof-of-work chain, which he can get by querying
network nodes until he's convinced he has the longest chain, and obtain the Merkle branch
linking the transaction to the block it's timestamped in. He can't check the transaction for
himself, but by linking it to a place in the chain, he can see that a network node has accepted it,
and blocks added after it further confirm the network has accepted.
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. One strategy to
protect against this would be to accept alerts from network nodes when they detect an invalid
block, prompting the user's software to download the full block and alerted transactions to
confirm the inconsistency. Businesses that receive frequent payments will probably still want to
run their own nodes for more independent security and quicker verification
Personally, i have a mobile wallet with few money in it. Like the same amount of money I would hold in my physical wallet.
if you are going to travel abroad, it may be handy to have a mobile with some money, as you can always find some place that accepts bitcoin and use it.