a spider core is a standard 3x3 core. it has 6 arms and the pieces rotate around a center piece. on a 4x4 spider core, the center piece (core) is hidden under the first layer, under the 4 center pieces (solvable layer). if you spread your cube apart and look in the centerpiece, there is typically a screw or a rivet. a spider core is used on the majority of newer puzzles (since the release of vcubes mechanism). a ball core is used on older 4x4 and 5x5 cubes. it is where there is a ball with grooves in which the pieces slide around in (like a rubiks 4x4). the problem with a ball core was that they were like storebought 3x3s. they are not adjustable. they have no springs in them so they lockup all the time and they do not cut corners that well. also, when the ball becomes loose, the cube becomes fairly unusable.
the dayan+mf8 has a unique design based off a hidden 2x2 screw spring mech (please correct if im wrong, i havent completely disassembled this puzzle personally) it is much like a lanlan 2x2 on the inside in which the pieces on the outside float around on. therefore, the inner layers of this cube work VERY well, however since the pieces are interlocked on the outside of the 2x2 mech, it makes the outer layer sluggish feeling and it has a strong tendency to pop. you did choose a very good cube, just be sure not to be so aggresive when you are heavily alternating between outer layer turns and slice turns. make sure the inner slices are aligned well before trying to perform a outer layer turn. it will pop if you force it ^_^.