The information you get will not change, the next step after getting it will still be the same. The only difference will be who tells you what you need to know.
Altho IIRC going to talk to Esbern at this point will also result in you getting a separate quest from the Blades that you can complete or not and the choice you make about it will have an affect on your future dealings with both them and the Greybeards. It won't affect your ability to complete the MQ though.