The answer is what is your aim. A big name will fetch you opportunities for your future, but a new name shall give you the zeal to create those opportunities for yourself.
Working with start-ups can be very interesting & a great learning experience in Nift, because they are starting up. You will be greatly involved in their management roles. Not only do you learn what you are hired for, but you get to learn a lot of other management strategies as the company is growing. You might have to work your sweat off as your employer too is doing so, but at each step, he will be kind enough to thank you for your efforts or atleast acknowledge them. Such people do not have pride & arrogance in them, because they are learning too as you are & if you are smart enough, they would be highly interested in using your skills to the best, which might look extra from the outside, but it is the building of your own foundation.
While a big name shall give you the comfort of a good set up & a guarantee of permanency & future commitments from the organisation, the expectations riding on you shall be high! On thing which they have in abundance is experience, which for them, you will always be new to. Here, your loss is your own because they make sure that you pay for the damages you make, which will make you smarter & a long-term thinker. You are basically in a fixed set-up, where what can distinguish you is the rise in profits. You learn that in a huge way. Because here they are stable, what is on for a toss is how much can you do good to them. Here, nobody cares about what you are doing to others, its only your immediate boss & your team who shall bother & then with the work created their bosses. Which means you won't be spared but you shall have no intervention too.
Both parts of the story are a learning experience, different for different individuals & unique in their own way, its about how you take & what you make of it.