Why did Hitler make exemptions for some Jews?

The answer to this question really depends on the time period, since Nazi policy toward Jews changed over time. In the beginning, German policy toward Jews was essentially to drive them from German borders. Many German Jews, including World War I veterans and people with designated "Aryan" spouses, were initially exempted from deportation to ghettos and camps. This was also true of the children of marriages between Jews and non-Jews.
This policy changed over time, however, and these people, including the elderly (who were also initially exempted from the pogroms) were caught up in the Final Solution. It should be noted that these exemptions only applied to German Jews, as well. Some Jews temporarily gained special treatment by cooperating with Nazis. These people, who included camp officials known as "kapos," very often died in the same camps as other Jewish men and women. Still others were spared immediate death in the murderous camps because their labor, and often their expertise, could be exploited by the Nazi state.
The very nature of the Final Solution, however, was that Jews were to be murdered, wiped out by the Nazis. No groups of Jewish people were exempted from this policy.
https://encyclopedia.ushmm.org/content/en/article/german-jews-during-the-holocaust

In general, the Nazis wanted to kill all Jews, but they were not able to. Many Jews lived in places outside of the reach of the Nazis like Britain, and they were safe. Others who lived in places of Nazi control were able to flee at the start of the war (and during it, as well). Jews fled to places like Sweden, which helped them avoid the concentration camp.
In terms of Jews in places of Nazi control, some were spared because they were hard workers and were not killed first. Those who were most capable of helping the Nazis were kept alive as they were free labor, which saved the Nazis a great deal of money. Eventually they would have been killed as well, but the war ended, and those who were able to survive that long escaped the grip of the Nazis.
Additionally, some Jews were allowed to live longer due to connections or making deals with SS officers or other Nazis. Also, another way of avoiding death was to have fake papers or something along those lines. A great and interesting read that will show some of this is the graphic novel by Art Spiegelman, Maus.