open-circuited stub VS stepped-impedance
Ignoring for a moment that that publication seems to be of very low quality (it reads almost like a student assignment, not like a paper/letter from a scientific publication, and at no point really provides any novelty).

The difference is simple: A stepped impedance is a section where the characteristic impedance changes. This impedance step causes reflections. Creating a short section of such a structure gives rise to resonances, and as a result many of these can be used to build a filter.

A stub is a transmissionline section that is connected to another transmission line. 

In the paper linked, the stub is the big piece that goes down, in the middle of the structure (the 11 mm long, 8 mm wide piece). 

It is an open-circuit stub because the end of it is not connected to anything - it is left open. By doing this, you generate a resonator again (as the waves reflect of the open-circuit end). 

In this paper, what they seem to be doing is create a filter from a lumped-circuit model, and then build the lumped elements with microstrip structures - the inductors are made with straight section of transmission line, the stub makes a capacitor.