The edges of any untrimmed NURBS surface exactly match a NURBS curve with the same degree, number of spans, knots and control points as the surface edge. If the NURBS surface is rational the matching NURBS curve will also have the same weights.
A curve on a surface with constant u or v parameter value is an isocurve. Edge curves of untrimmed surfaces are isocurves. Isocurves are NURBS curves with the same degree, number of spans and knots as the surface in the direction of the isocurve. However control points of interior isocurves only match the control points of the surface in special situations.
A curve on a surface which is not an isocurve may or may not exactly match a NURBS curve. A curve on a surface can be approximated to any desired level of accuracy by a NURBS curve but the number of spans, knots and control points will usually increase as the accuracy becomes tighter.
Trimmed surfaces are more complicated and will be discussed in another post.
A matching edge curve can be created using DupEdge and selecting the edges to be duplicated. The entire set of edge curves can be created as a polycurve using DupBorder.
For a discussion of Bezier surfaces and NURBS surfaces see Bezier, Single Span & NURBS Surfaces - Differences and Similarities