Why is it more important to check for errors during DNA replications than during protein synthesis?

Because if an error is made during protein synthesis, the result is at worst one bad protein. If an error is made during DNA replication, that error will persist for the entire lifespan of that cell and be inherited by every daughter cell it produces. Should the error prove deleterious, the effects can be catastrophic for the cell or the organism. For example, if an error occurs in a protein coding segment of DNA, it is possible that every protein that locus generates will now be defective.