Short answer: BGP Weight is a local path-selection value used by Cisco-style BGP implementations to make one router prefer one BGP path over another. Higher Weight wins, Cisco documents it as the first best-path comparison step, and Cisco also states that Weight is local to the router where it is configured.[1][2]
Use Weight when one router needs a local outbound preference. Use LOCAL_PREF when the whole autonomous system should agree on an outbound exit. Do not use Weight when you expect a policy to propagate through iBGP, influence inbound Internet traffic, or become a portable standards-based BGP attribute.