These are both cognitive errors, but conservatism is a belief perserverance bias and Anchoring is a data processing bias.
I think of the difference this way:
For conservatism you preserve your original belief by putting too little wieght on new information(hence it is a belief preservation cognitive bias)
For Anchoring you have processed data incorrectly and put more (undue) weight on the original information (hence a data processing cognitive bias)
the difference has to do with what information you are weighting. Correct weight on original data, but too little weight on new data = conservatism bias. Too high of weight on original data, but new data is weighted correctly = anchoring bias.