I'll start with my simple definitions. For me, authoritative leaders lead by position; servant leaders lead by serving.
All leadership is influence. How you lead determines how you exert influence. Let's look at some examples.
In one church, there is a clearly defined group of elders. The culture of the church is clear that the elders are the "rulers" of the church. Decisions are made by them and any idea or project that someone wants to undertake must first be run by the elders for their "approval." Those who bypass this system, however genuinely, are directed to submit to the authority of the elders. For example, one church member started a home bible study. When the elders of the church heard, they called the church member into their meeting, demanding an explanation of why this was started. They explained that they, the elders, disagreed with home groups because they split the church away from the communal structure they desired (worship and one all-ages bible class). They told the member the home group would need to be disbanded. The structure of the church, seen through this decision, is clear--it is top-down leadership all the way. People submit more out of timidity and fear than out of love and respect. This is authoritative leadership.
This is an extreme example, to be sure. Not every authoritative leadership group is as rigid as in my example. But authoritative leaders who "lead" based on their positions exert authority more than influence. In fact, we might argue that their influence is negative, because it promotes fear and anxiety more than love and respect.
Another church operates differently. This church also has a group of elders but they call themselves "shepherds" and focus on relationships more than meetings and decisions. In fact, these shepherds encourage church members to be creative in ministry applications. In the course of visiting the church members, one member mentioned to them an idea to start a food pantry for needy families in the neighborhood, these shepherds commended this member for their creative thinking, offered to create space in the church building for the food pantry, and committed themselves to serving in it as a way to serve this member and the needy families who would be served. This is servant leadership.
In short, authoritative leaders use their "authority" (usually derived from a position of leadership) to make decisions, delegate, and dictate from the top-down. Authoritative leaders often struggle to relate to people.
Servant leaders have authority, but it's granted to them through their serving of others. These leaders exert real influence over others through their relationships. Relationships are key for servant leaders.