The 200-600 is heavy. If you're not used to the weight, it can be daunting over a period of time. Since the lenses are the same in speed, you aren't gaining any low-light performance. But as Pieter suggests, what about the short end? Is 200 going to be a problem? Are you shooting from the sidelines or from the stands? If after reviewing your most commonly used focal length you find 200 won't do it, then you could look at the Sigma 60-600 Sport. It's about the same price as the Sony and would give you more at the wide end but is a bit heavier yet.  Two other options would be the Sigma 150-600 and the Tamron 150-500. Both are similar in weight to the others, maybe a touch lighter.   Lastly, and if reach is your problem, then this may not work, you could look at the Tamron 50-400. It's an amazing lens, sharp from end to end, considerably smaller and lighter than the others, and costs less. I use it all the time, thousands of shots, and it never disappoints. But 400 may not be long enough for you.  FWIW, I have the 200-600. It gets taken out about 3 times a year, while the 50-400 goes with me every time. If you go with one of the larger lenses, plan on using a monopod just to help with supporting it for a few hours.     

Some questions to help you formulate the answer to your own question: In what sense do you feel your 70-350 is lacking? What do you hope to gain from this 'upgrade'? What % of your shots are in the 70-200 range? If this % is significant, the 200-600 is not an option unless you carry two cameras or don't mind lens swapping. What % of your shots is at 350mm and potentially could have benefitted from more reach? Do you need high resolution shots or is cropping in post an option to fix your compositions? Would carrying a lens over twice as big and over 3 times as heavy have a negative impact on your photography (think of fun, mobility, physical strain, packing volume)?

Im looking to upgrade my Lens for high school football games, I currently have the Sony - E 70-350mm F4.5-6.3 G OSS Telephoto Zoom Lens for E-mount.   I am comparing the Sony e70-350 telephoto zoom lense to Sony 200mm-600mm G OSS.   Help and suggestions please!