1) You're measuring protein expression dynamics; maturation must be fast relative to the timescale of what you want to observe.
2) You need a high fraction of fluorescent protein to be mature; maturation must be fast relative to protein degradation and cell growth rates.
For point #2, maturation be both fast and complete (little misfolding or other problems leading to no fluorescence). All else equal, faster maturation gives better signal:background/noise, but all else is rarely equal so the fastest maturing FP is not necessarily the best choice.
1) You're measuring protein expression dynamics; maturation must be fast relative to the timescale of what you want to observe.
2) You need a high fraction of fluorescent protein to be mature; maturation must be fast relative to protein degradation and cell growth rates.
For point #2, maturation be both fast and complete (little misfolding or other problems leading to no fluorescence). All else equal, faster maturation gives better signal:background/noise, but all else is rarely equal so the fastest maturing FP is not necessarily the best choice.
Rapidly maturing FP fused with slowly maturing FP (with non overlapping excitation / emission) can serve as a 'tandem fluorescent timer' that allows to monitor protein stability (as described by A Khmelinskii et al. http://www.nature.com/nbt/journal/v30/n7/full/nbt.2281.html )
Remember that most FPs´ maturation times exceed speed of embryonic development, particularly when using live imaging. Expect a lag of several hours! It may be smart to detect mRNA transcripts of the reporter transgene by ISH instead (shorter half-life of RNA compared to protein), or reveal transgene expression by IF (antibody can detect FP before the actual FP complex is formed). Drawback however is loss of live situation.
It is particularly important if you want to develop expression reporters. In bacteria, because of the fast replication rate and bacterial adaptation, it can rapidly become an issue. Using an antibody can indeed solve part of the problem, but you loose the capacity to do live experiments. I would say that picking the faster maturing fluorescent protein from the start can resolve several downstream problems. The best variant for GFP are: GFPmut2, GFPmut3 (half-time for maturation approx. 5 min); for RFP, mKates2 should be better than mCherry (h-t for maturation 40 vs 20 min), but both are less good than GFP.
I would say that the stability of the fluorescent protein is also very important to consider. Less stable GFP will mean less signal but better sensitivity in the response.
In those two papers plus the last one from another group, you will find some useful information. Good luck
For all normal studies, you do not have to worry about its maturation time. Well, if you are working on the thermodynamics of these proteins (especially) and that too some critical enzymes, then the maturation time will be a factor needed during computaion of your results.