They're the two ways to fix the rotor to the hub. 6-bolt: six M5 Torx bolts on a 44 mm circle, tightened in a star — universal and serviceable in any shop. Center Lock: the rotor engages the hub splines and is fixed with a central lockring at 40 N·m — faster to fit and self-centering, but a specific hub interface. Your hub decides it: 6 holes or splines. You can cross from Center Lock to 6-bolt with an adapter, not the reverse.
Your hub decides this, not your brake: six little bolts or one central nut? That determines which rotor you buy.
Look at your hub: if the rotor is held by six bolts, it's 6-bolt (universal, cheap, reliable); if it sits on splines with a central nut, it's Center Lock (fast, centered, a bit pricier with its tool). In performance they're equivalent; the choice is set by the hub you already have. Center Lock needs the lockring tool (cassette or BB type depending on the lockring).
A 6-hole hub uses a 6-bolt rotor; a Center Lock hub uses a Center Lock rotor (open Shimano/SRAM standard). You can mount a 6-bolt rotor on a Center Lock hub with an adapter; a Center Lock rotor can't be adapted to a 6-hole hub. The mounting type doesn't change the size or the caliper mount.
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In braking they're equivalent. Center Lock is faster to fit and self-centers; 6-bolt is universal and cheap. Choose by the hub you already have.
Yes, with a Center Lock-to-6-bolt adapter. Not the reverse: a Center Lock rotor won't fit a 6-hole hub.
The lockring tool: the cassette one (FR-5) for the internal lockring, or the BB one for the external (required on 15/20 mm axles).