I think Bezier Curve by default creates a curve by combining several curves with SplineDegree 3. So, you can get your two plots to match by setting the SplineDegree of the BezierCurve to the number of control points.

But that would be getting your BezierCurve to match your BezierFunction, and you were asking for the other way. In that case, you'd have to replicate BezierCurve's default behavior. You'd need to create sever BezierFunctions by taking your control points 3 at a time. If you wanted to have one single function with parameter running from 0 to 1, you'd have to scale each individual BezierFunction to the appropriate part of the parameter's domain and compose them all together.

Or do it yourself, the construction is quite easy:

Clear[Smoothed, CreateGraphics]
Smoothed[pts_List, \[Alpha]_?
   NumericQ] := {1 - \[Alpha], \[Alpha]}.# & /@ Partition[pts, 2, 1]
CreateGraphics[pts_List, \[Alpha]_] := 
 Module[{sublines, smooth, \[Theta], gr},
  sublines = NestList[Smoothed[#, \[Alpha]] &, pts, Length[pts] - 2];
  smooth = Nest[Smoothed[#, \[Theta]] &, pts, Length[pts] - 1];
  sublines = 
   Riffle[Line /@ sublines, Hue /@ Range[0, 1, 1/(Length[pts] - 1)]];
  gr = ParametricPlot[smooth, {\[Theta], 0, 1}];
  Show[{Graphics[{sublines, PointSize[Large], 
      Point[smooth /. \[Theta] -> \[Alpha]]}], gr}, 
   PlotRange -> {{-2, 3}, {-1, 2}}]
  ]

Manipulate[
 CreateGraphics[
  pt, \[Alpha]], {{pt, {{-1, 0}, {-0.8, 1.6}, {0.8, 1.8}, {1.8, 
     1.2}, {2, 0}, {1.4, -0.8}}}, Locator, 
  LocatorAutoCreate -> {3, 8}}, {{\[Alpha], 0.3}, 0, 1}]

giving:

Doesn't this do exactly the same thing as BezierFunction? I mean, it is still necessary to stitch together multiple degree-3 parts with this method too.

Maybe a bug, maybe some accuracy thing (though I tried AccuracyGoal, PrecisionGoal, MaxCellMeasure et cetera to improve but to no avail).

pt = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {2, 0}};
g = Graphics[{BezierCurve[pt]}];
Show[DiscretizeGraphics[g], g]

Maybe also something to do with the default SplineDegree....

Indeed! Good hunting! Now, it would be nice is there was a @mathematicabugreport account. So you could just submit this by mentioning it....