Extradition is done for specific charges. A principle found in virtually all extradition treaties called the "rule of specialty" says that the country requesting extradition may not prosecute the defendant for any crimes except the ones for which extradition was granted without the permission of the extraditing country, except for crimes committed after the defendant is extradited. This protection expires once the defendant has been released from jail and had a fair chance to leave the country.
The rule of specialty doesn't necessarily mean the other charge needs to be dropped, but the defendant can't be tried for it as long as the rule applies. If they're later in the country for another reason (or don't leave when they have the chance), they can potentially be rearrested for the other crime. But as long as they're only in the country because they were extradited, they can't be tried for any other past crimes without the extraditing country's permission.