If you are considering applying for asylum and your home country has been designated for Temporary Protected Status (TPS), you can apply for either or both, but you should act quickly and TPS should be your top priority.
You should apply for TPS as soon possible as long as you meet the eligibility requirements, described in Temporary Protected Status (TPS): Who is Eligible?
Once your TPS application is received by U.S. Citizenship and Immigration Services (USCIS), your time spent in the U.S. will no longer be considered illegal. That will allow you to live free of the fear that you will have a run-in with immigration or other law enforcement authorities. It will also help you if you ever become eligible for a U.S. green card based on a category other than asylum. For most types of green cards, accruing unlawful presence of six months or more in the U.S. leads to inadmissibility issues. But if you apply for TPS within 180 days (or approximately six months) of your visa expiration, you will not be subject to the three-year time bar for unlawful presence in the United States.
Although TPS is a temporary immigration status, it also will allow you to legally obtain employment in the United States. Applicants for asylum are not allowed to work immediately after submitting their application to USCIS unless they already have a valid work permit. They will be allowed to work only after their asylum application has been approved or they've waited 365 days or more with no decision or outstanding, applicant-caused delays on their asylum application.
They must also meet the other eligibility criteria described in How to Apply for a Work Permit While Awaiting an Asylum Decision.
Many TPS recipients are also eligible for asylum due to political turmoil in their home countries that causes them to fear persecution based on race, nationality, religion, social group or political opinion. Others, however, cannot show that they have a personal reason to fear persecution—they were just caught in the middle of a dangerous situation. If you can make a good case for asylum, you should request this as well, as it leads to greater long-term benefits.
Unfortunately, you are also under a time crunch when applying for asylum. Ideally, you should apply before the one-year anniversary of your arrival to the United States. Otherwise, you will have a higher burden of proof when convincing USCIS to approve your asylum application, and you might not be eligible for a work permit.
If granted asylum, you can apply for permanent residence (a "green card") one year after the date of approval. This will give you significantly more benefits than TPS.
To learn more about asylum, visit Applying for Asylum Status.