It depends on your location. A felony conviction can limit your rights in various ways, and those rights may or may not be restored at the state level. See this article for discussion of federal convictions and collateral civil disabilities. The Dept. of Justics says that "a presidential pardon will restore various rights lost as a result of the pardoned offense and should lessen to some extent the stigma arising from a conviction, it will not erase or expunge the record of your conviction", and "most civil disabilities attendant upon a federal felony conviction, such as loss of the right to vote and hold state public office, are imposed by state rather than federal law, and also may be removed by state action".
This article surveys civil disabilities of convicts on a state by state basis, as well as at the federal level. For example, voting rights are set at the state level, so a state has to have statutes restoring rights upon a pardon if you are to get your voting rights back. Service on a federal jury would be restored per 28 USC 1865(b)(5); the federal felon in possession crime has an exception encoded in it in the definition of "crime punishable by imprisonment for a term exceeding one year" which says
What constitutes a conviction of such a crime shall be determined in
accordance with the law of the jurisdiction in which the proceedings
Any conviction which has been expunged, or set aside or for which a
person has been pardoned or has had civil rights restored shall not be
considered a conviction for purposes of this chapter, unless such
pardon, expungement, or restoration of civil rights expressly provides
that the person may not ship, transport, possess, or receive firearms.
So if you are a state felon, you need a state pardon (or other state procedure) and if you are a federal felon, a state pardon does no good, you need a federal pardon. However, 21 USC 862 makes mandatory the ineligibility for federal benefits after a third conviction for drug trafficking (either state of federal law), and there is no "rights restoration" clause that restores such rights after a pardon even if the convictions are all federal. I have not located any case law establishing whether federal benefits rights restoration for federal convictions flows automatically from a federal pardon (not many 3-time drug dealers get federal pardons). A federal felony conviction can also be used as a factor in granting a security clearance (applicable to certain jobs), and current law does not say that a pardoned conviction cannot be considered in deciding on a clearance.