With the select max(id) + 1 approach, two sessions inserting simultaneously will see the same current max ID from the table, and both insert the same new ID value. The only way to use this safely is to lock the table before starting the transaction, which is painful and serialises the transactions. (And as Stijn points out, values can be reused if the highest record is deleted). Basically, never use this approach. (There may very occasionally be a compelling reason to do so, but I’m not sure I’ve ever seen one).
The sequence guarantees that the two sessions will get different values, and no serialisation is needed. It will perform better and be safer, easier to code and easier to maintain.
The only way you can get duplicate errors using the sequence is if records already exist in the table with IDs above the sequence value, or if something is still inserting records without using the sequence. So if you had an existing table with manually entered IDs, say 1 to 10, and you created a sequence with a default start-with value of 1, the first insert using the sequence would try to insert an ID of 1 – which already exists. After trying that 10 times the sequence would give you 11, which would work. If you then used the max-ID approach to do the next insert that would use 12, but the sequence would still be on 11 and would also give you 12 next time you called nextval.
The sequence and table are not related. The sequence is not automatically updated if a manually-generated ID value is inserted into the table, so the two approaches don’t mix. (Among other things, the same sequence can be used to generate IDs for multiple tables, as mentioned in the docs).
If you’re changing from a manual approach to a sequence approach, you need to make sure the sequence is created with a start-with value that is higher than all existing IDs in the table, and that everything that does an insert uses the sequence only in the future.