What is the difference between min- cut formulation and (bi) partitioning formulation?

Roughly speaking, in minimum cut problems, the goal is generally to find a minimum cut (possibly weighted) between two fixed sets of vertices, called the sources and the sinks. Given one source and one sink in input, the problem can be solved in polynomial time, a famous theorem in combinatorial optimization.

On the other hand, in graph partitioning problems, the goal is generally to find a partition of the vertices into subsets with a maximum size while minimizing the cut (possibly weighted) between these subsets. For example, the bisection problem consists of finding a partition of the vertices into two subsets of equal size, while the minimum cut between the two subsets is minimum. This problem is known to be NP-hard.

But in the end, this is a matter of precise definition of the problem to be solved. Some cut problems are polynomial; some others are NP-hard (for example, the so-called maximum cut problem). The same holds for graph partitioning problems.