A 243 will shoot flatter than a 30-06.
The heavier 30-06 bullet will likely be less effected by wind and likely be more accurate at longer range and certainly have more power when hunting larger game. Small differences in how flat cartridges shoot are highly over rated. If you know the range anyone can compensate for bullet drop. Anyone shooting over 300 yards should be using a rangefinder and using a scope with either a long range reticle, or adjustable turrets ayway.
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I've got a Savage .243, and it shoots the flattest trajectory I've ever seen,
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There are sevearal cartridges that shoot flatter than a 243, although it is one of the better ones.