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Pro True desktop browser experience
In flash mode you get remote access to desktop Firefox on a Linux server. Unbeatable desktop web compatibility on Android.
Con Flash mode has ads
You can also get an annual subscription to turn these off, but it's $10/year. That's kind of pricey when you can get Puffin Browser Pro with unlimited service for a single payment of $4.
Con Videos are compressed twice
In Flash mode, you get an interactive streaming video of a desktop browser running on their server, but this means that when you watch videos (a major use case for Flash mode), which are already compressed, it has to be lossy compressed again to get to you, with noticeable loss of quality. So it's better to watch videos with Flash mode off if you can. There's also a bit of lag when scrolling and such.