Assuming you mean the thickness of the casing on the tire....
Given a similar tread pattern:
-Thin casing will be lighter and sometimes more supple, but will be very vulnerable to cuts, pinch flats, thorn punctures.
-Thicker casing will generally be a more durable tire that resists pinch flats, tears, and punctures better. The biggest disadvantage would be weight and sometimes ride quality (not as supple, though often can be run at much lower pressures to compensate for this).
if you are refering to tire width, thin tires create less rolling resistance and are obviously a bit lighter, making it feel easier to pedal (as there is less resistance). Wider tires can absorb bumps a bit better (if properly inflated) and due to the added contact area, can provide more traction.
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