Firstly I don't know about the implementation details, so maybe I have some misunderstanding. However I still think I can help. Correct me if I'm wrong :-D.
BIP37 protocol allows the full node to find out relevant transactions from the blockchain ledger with bloom filters, so that the lightweight client would not have to download the full blocks. However it's still very inefficient. It's much less efficient than a fully indexed server like ElectrumX, Electrs etc. The full node still needs to scan the blockchain, which requires reading the hard drive intensively.
Note that Electrum protocol (essentially) requires the lightweight client to send its addresses directly, so that its privacy is not good.
BIP157/158 (aka Neutrino) protocol let the full node build a block filter index in advance, so that a lightweight client only needs to download the block filter index only firstly. With such block fitler index, the lightweight client can scan through the filters to find out which blocks may be relevant, and then it can download those possibly relevant blocks to scan out the relevant transactions from them.
Neutrino protocol is still not as efficient as a fully indexed server, like an Electrum server (or an online block explorer), however it's efficient enough to allow an ordinary laptop to scan though the whole blockchain in only 1-2 minutes, which is almostly two orders of magitudes more efficient than directly scanning the blockchain itself. The block filter index also takes much lower disk space than a fully indexed server. The bandwith consumption is obviously higher because the block filter index currently takes ~5GB, however once it's completely downloaded, it can be then reused freely.
However, Andreas Schildbach, maintainer of bitcoinj and the bitcoin-wallet for Android, thinks that Neutrino still has some limitations/problems, so that it's not really an alternative of BIP37.
By the way, what a Merkle proof can do is very much limited than what a lot of people supposed.
Merkle path proves nothing but the existence of the transaction data itself, which is essentially just a timestamp. This fact essentially makes the "reclaiming disk space" mentioned in Satoshi' s whitepaper unrealistic.
SPV light clients are facing similar problem, that even Satoshi himself also mentioned that SPV "can be fooled by an attacker's fabricated transactions", while unfortunately the countermeasure (aka "fraud proofs" nowadays) which Satoshi assumed seems impossible (or always incomplete) in reality.
