Compare fee : Bech32 & P2SH

[hugeblack]

Technically, is there any difference between Bech32 "BIP_0173" and P2SH "Pay to script hash/transactions to be sent to a script hash" segwit Transactions?
Why do some say that the transactions fee using  Bech32 will be cheaper than P2SH?

[LoyceV]

A lazy quote as answer:

A typical transaction of one input, two outputs is 226 bytes legacy, 166 vbytes P2SH-P2WPKH segwit (27% less), 141 vbytes pure bech32 P2WPKH segwit (38% less).

If you have many inputs, the savings are larger (and less for many outputs). For 10 input, 1 output legacy is 1515 bytes, 953 vbytes P2SH-P2WPKH (37% less), and 722 vbytes P2WPKH (52% less).

Multisignature transactions benefit even more. Multiinput 2 of 3 multisignature transactions in bech32 P2WPKH segwit are 75% smaller.