OEI 5 min TO ppwer vs 10 min TO power
 

I'm trying to understand the performance differences between a business jet who is certified for One Engine Inoperative (OEI) take off power of 5 minutes vs the same model whose engines are certified for OEI TO power of 10 minutes. If a SID departure requires a climb gradient of 400 feet per NM, shouldn't either aircraft have the same weight limits? From what I've seen in the performance charts, the 10 minute aircraft can go with a slightly higher weight. This just doesn't make sense to me so I must be missing something. Thank you.
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In general, 10 minute takeoff is needed when there are more distant obstacles to clear (e.g. more than 5 minutes but less than 10 minutes at engine-out speed, or a relatively tight turn is required that would be challenging with an engine out).

Used to fly the same type which had both 5 and 10 minutes on the fleet. Some frames were 5 minutes, others 10. As the engines all got swapped around, it was all pretty much nonsense and was driven by the tail number rather than anything to do with the engines.

10 minute Takeoff is an 'extra cost' option from the engine manufacturer - mainly because it has warrantee implications as 10 minutes at Max Takeoff takes a lot of life out of the engine. You are correct that the engine doesn't change (same thing is true for most ratings changes - the engine company charges for because higher ratings reduce engine life and have warrantee implications. Also because higher ratings are 'value added' to the customer so they expect to get paid for that extra value (ditto for 10 minute takeoff).
The airframer also has to make some changes to EICAS (or the equivalent) as the takeoff EGT (and sometimes rotor speed limits) are higher for TO than Max Con, so the exceedance logic needs to reflect that.