Is training on saved weights the same as starting over with larger dataset?

Part 1 (2017)
1

I have a scenario I have been wondering about.

If I save weights for a trained data set with 1 million things (images, audio, text, whatever) and then add another 1 million things and train further. Will I get similar results if I took 2 million things and trained a model? Would either perform better?

What I am trying to understand is how would you handle a model for something like anti-spam where it is updated around the clock but is also used to predict around the clock.

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@dradientgescent Wondering if you were able to figure this out. I have a similar situation wherein the new data arrives on a daily basis and what should we do to accommodate the new data as appending to old dataset and running takes long training times.