In a strange quirk of the Texan political system, the state constitution stipulates that the governor of Texas cannot be governor if they are not physically present in the state. So if the governor is out of the state for any length of time, then the lieutenant (lt.) governor effectively takes over. This provision of the Texas state constitution gives the lt. governor a remarkable degree of power, though only for a relatively short period of time. If the governor should die in office, then the lt. governor will automatically take over.
It should be remembered that the lt. governor doesn't derive their authority from the Texas constitution but from the substantive powers awarded to them by the state legislature. These powers confer on the recipient a wide degree of discretion, which means that the office of lt. governor can be pretty much what the holder wants it to be. Lt. governors can choose to be active or passive, but in most cases it's the former, as the office, as well as being powerful in its own right, can also be a springboard to higher things.
It's in the lt. governor's interactions with the state senate that their powers can be seen at their fullest extent. The lt. governor appoints all the committee chairs of the committees in the senate, determines where the bills are going to be sent, and to what committees and the timing. Every bill has to pass through committee before it ends up on the floor of the senate for a vote. And it's the lt. governor who decides which committee to send a bill to, and this gives them extraordinary power.
Even when a bill finally makes it out of committee, the lt. governor gets to decide when it's voted on. So at every stage of the legislative process, the lt. governor enjoys power and influence.
However, it should be stressed that because such powers are conventional rather than constitutional, they can easily be taken away. In that sense, one could say that the lt. governor, though substantively more powerful than the governor of Texas, is at the same time weaker in the formal sense.