The time signature 4/4 means: each bar has 4 beats, and those beats are crotchets (quarter-notes). The strong beat is on 1, a medium beat is on 3, and beats 2 and 4 are both weak:
1 2 3 4
S w M w
The time signature 2/2 means: each bar has only 2 beats, and those beat are minims (half-notes). The strong beat is still on 1, and 2 is a weak beat:
NB: If the relative strengths of these beats is not apparent in some way, the time signature is largely meaningless.
Count through the two patterns above out-loud, with significantly more emphasis on the strong, and significantly less on the weak beats.
In a simple rock-style groove: the bass drum goes on strong and medium beats, and the snare goes on the weak (back)beats. The hi-hat pulses both on and between beats. Hitting the crash cymbal on 1 helps mark this as a strong beat.
In the example below, the tempo indication changes from crotchet=100 to minim=50 (which is in some way equivalent). This halves the number of beats per minute (BMP) to maintain the duration of each note value.
Still, the difference between a single bar of 4/4 and two bars of 2/2 is somewhat subtle (two bars of 2/2 can sound like a slow bar of 4/4). It becomes more obvious when four bar phrases are considered. In the absence of other instruments (that say, play a chord every bar) adding fills solidifies the four bar phrase structure, and hence the "feeling" of the time signature.
To play-up the slower rate of beats, it also helps to keep the hi-hat pulses continuing with quavers. You can change the hi-hat pulses to crotchets (as above), but you'd have to be sure to really emphasise every beat 1 to be convincingly in 2/2 (unlike 4/4: every kick should get the same amount of strong emphasis in 2/2).
Play a few repeats of each of the four bar phrases above on a drum kit (while counting in your head) and loop back the beginning without pausing in-between. Do this until 2/2 really feels like a different time signature. Then, try with crotchets on the hi-hat for the 2/2 section; it'll be harder to make it feel like four bar phrases of 2/2, but it is certainly possible if you get the relative emphasis right.