The flowers that say rare, are rare, regardless of whether or not they say they are. For whatever reason, glitch maybe, it just doesn't show it. There are different charts online that show you which are more rare to produce. Black, blue, and purple seeds are rare. So to answer your question, no, there is no difference.
I noticed that "Rare!" only shows up when cross-breeding flowers of lower rarity. For example, when I cross breed with my planted blue tulip, blue seeds don't have the "Rare!" icon. In fact since blue is the rarest, any colors that result from breeding with blue don't show the icon. The same occurs when I breed orange x pink and get pink seeds etc. But if I get a black/purple seed from orange x pink it shows up as "Rare!"
When the "Rare!" pops up, it's because you got a mutation.
red/white mutates to pink/white 70% of the time
white/white mutates to black/white 70% of the time
pink/black mutates to purple/blue 70% of the time
yellow/yellow mutates to coral/white 70% of the time
coral/coral mutates to blue/white 60% of the time
red/blue mutates to purple/white 30% of the time
If you cross orange tulips purchased from Lliod, they are orange/white (orange is dominant over the white). Crossing those, you'll get:
orange/orange 25% (failure rate 10%), so 0.25*0.9 = 0.225 (22.5%)
orange/white 50% (failure rate 10%), so 0.5*0.9 = 0.45 (45%)
white/white 25%, but only 30% stay white/white (failure rate 10%), so 0.25*0.9*0.3 = 0.0675 (6.75%)
white/white 25%, but 70% mutate to black/white (failure rate 10%), so 0.25*0.9*0.7 = 0.1575 (15.75%)
If you crossed orange/white and orange/black, you would get black/white without mutating and so it won't be marked as rare.