When there is no spend transaction the search method is:

    Next Private Key -> Public Key -> Hash -> Bitcoin Address -> Compare Bitcoin Address, repeat until found

    Note that due to the hashing functions used the Bitcoin Address match is expected within a private key range of only 2¹⁶⁰

A spend transaction exposes the public key so when there is a spend transaction the search method is:

    Next Private Key -> Public Key -> Compare Public Key, repeat until found

    Note that the private key range in this case is the full 2²⁵⁶ but there are ways to speed up the process so the effective security is reduced to only 128 bits.

See the following:


So the two situations are:

     Full entropy 256 bit private keys with multiple spend transactions have 128 bits of security

     160 bit Bitcoin address from a full entropy 256 bit private key with no spend transactions have 160 bits of security

So Bitcoins kept on a Bitcoin address with no spend transactions are safer (160 bits of security) than Bitcoins kept on a Bitcoin address that has spend transactions (only 128 bits of security).