If a world existed in which, by any mechanism you like, the world were constantly exposed to sunlight or visible light of comparable brightness on all sides (so no side is correspondingly in permanent darkness), how would Flora and Fauna have evolved differently to take advantage of this fact?
So, assuming all day, all the time, and ignoring what caused this...
First you have to consider the Circadian rhythm. This is an evolutionary trait developed by all animals to take advantage of the day night cycle. For example, Plants store energy during the day, and burn it at night to keep growing (science).
With no night cycle, There is no need to keep track of time (unless there are still seasons). So all nocturnal creatures would become isolated to caves (and things like bats would have to evolve to be effective in day and caves.) Plants wouldn't need a surplus of energy, so they could focus 24-7 on raw growth (so they will grow faster)
Animals would also evolve to minimize rest needed (though some animals will still sleep 16 hours a day. Lazy koalas)
Since there wouldn't be a 'cooling' side of the planet anymore though, So Earth is going to have to evolve a balance between energy gained/radiated.
(Community wiki - please feel free to modify/improve this answer)