Ask the next person you see on the street if they know the answer to that question, and chances are they will not. Both of these state government positions are incredibly powerful and important. What exactly is the answer? Addressing it is complex for a number of reasons.
State vs federal government
The first thing to clarify is how the United States has a unique status in the way the federal government's power is restricted and kept in check by state governments. This was intentionally done, as the founding fathers feared a central government obtaining too much power. In simple terms, our constitution outlines what the federal government can do while everything else is reserved for the states. For example, the federal government was given the authority to oversee foreign policy, make as well as veto laws, modify conflicting laws between states, and other such powers. State governments are free to establish school systems, conduct elections, and anything else the Federal government cannot do.
Our political party systems adhere to this difference as well. Both Democratic and Republican national committees head fundraising and campaign activities; however, this is primarily for presidential campaigns. In addition, they cannot direct the activities of members of the party. The structure of both parties varies from state to state based on state laws as well as party rules on the state and national level.
A basic way to look at the governorship is seeing it as akin to being a "president" of a respective state. This position is part of the executive branch, answering only to the president. Enforcing laws, pardoning criminals, and appointing people to positions are all powers a governor can avail him/herself of.
Senators, on the other hand, are members of the legislative branch of government. What that means is they are responsible for representing their constituents through making and voting on laws. Many of them serve in multiple committees, which wield a large influence on major issues throughout the country. It is also important to recognize that there are state senators as well as the senators in the U.S. Congress.
And the winner is…?
Back to the original question: Who holds more power, a senator or a governor? The answer is that this is a trick question: Neither one has more power than the other. Remember that we have three branches of government designed to keep each other in check. Senators make laws, which the governors must then enforce. Yet this does not mean a senator outranks a governor. While a governor is the head of his or her state, they have no authority over a senator; the only right given to governors is appointing a replacement senator if they leave office for any reason.
Typically, senators are involved in representing their state's residents on a national level, while a governor operates on the state level. Note that there are governors who have more prestige than some senators, and the opposite can be true as well. (It all simply depends on what state they are from, their seniority, who they know, etc.) Although confusing and somewhat ambiguous, this answer is vital in understanding why our government functions the way it does.