There is no relation between private keys which have similar characters, public keys which have similar characters, addresses which have similar characters, or any subset thereof.
Here are two random private keys which differ by only 1:
02D615E2968401894AB0704E3B71E3C125F9B9A0C7ED3EDC204E1A0954B3E5C0
02D615E2968401894AB0704E3B71E3C125F9B9A0C7ED3EDC204E1A0954B3E5C1
And here are the two compressed public keys they generate:
029b438064a9e91a868390ed5cb9a66a6141141bc365f9cc064a8d1313426aa7a6
033bc0203bf01ccd77477a14aded37b489ca66d959bec40b90b3ec408a95a3db0c
As you can see, even a tiny different in the private keys results in such a difference in the public keys that they might as well be any two public keys - there is no clear link between them whatsoever.
Now, let's take the first public key from above, and a second public key which is simply n+1 of that public key:
029b438064a9e91a868390ed5cb9a66a6141141bc365f9cc064a8d1313426aa7a6
029b438064a9e91a868390ed5cb9a66a6141141bc365f9cc064a8d1313426aa7a7
And let's derive the two addresses these public keys generate:
1A4dJjJH3LYX6HZwujjQXCWpni2p3ZNRSr
1NUhDqSFqwkLFcZfdaCaB366J5KY7b842n
So again, as you can see, a tiny change in the public key results in two addresses which are completely different.
And the reverse is similarly true - any addresses which do happen to have several characters the same will have public and private keys with no obvious relation to each other.