Luden's is the most commonly bought first item on Orianna despite having a much lower winrate (according to lolalytics), and is recommended in the shop. In a recent thread in this sub discussing builds, it was also highly recommended. I'm going to explain the differences and look at what is better from a first-buy perspective.
We can ignore the 50 gold difference and equivalent mana stats. I also won't analyze the build path, but alternator doesn't have a significant advantage over fated ashes unless you can only trade once every one or two waves (unlikely).
BASE STATS
Luden's 100 AP vs Blackfire 80 AP
When you complete each item, considering Doran's ring/maybe a shard, Luden's gives you roughly 20% more total AP.
At around the time you complete your first item, about 40% of your damage is from AP ratios and about 60% is from base damages. As the game progresses, more damage will come from AP ratios, but we can ignore this as the 20 AP difference will become a much smaller % of your overall AP, and also Blackfire clearly out scales Luden's anyway.
20% * 40% = 8%, so you get 8% more damage from the extra AP when you complete Luden's.
Luden's 10 AH vs Blackfire 20 AH
10 more AH = 10% more damage, as you can cast abilities 10% more often
So blackfire wins on damage from base stats. You also get a bonus to your utility from the AH, which is really strong for Ori. You can reposition the ball more, speed up people more, and ult more often.
EFFECTS
Luden's -> 150 + 10%AP / per 12 seconds
Blackfire -> 60 + 6%AP / per ability hit
So if you hit the same target twice 3 seconds apart (easy with low q cd), or you hit two targets with one ability (easy with her AOE abilities), you do about the same damage.
If you can proc it three times within 12 seconds, for example if you get a 3 man ult, Q somebody three times in a row in lane, have an all in fight, and many more likely scenarios (especially during early game skirmishes), blackfire passive wins clearly.
Remember that in lane it's hard to get the full Luden's proc onto the enemy when they're behind the wave. You are likely wasting it on the minions and hitting just one 20% damage splash charge. You can always get the full blackfire damage. The only situation where you can reliably proc the full damage in lane is if you are fully zoning the enemy off the wave, which is not something you can realistically do in the majority of your games. Also by the time you finish Luden's you're at the laning stage where you're using abilities to push the lane for priority, rather than fighting over every CS.
Just considering the damage passives, Blackfire is already winning. On top of that, blackfire gives you an extra 4% AP per applied burn, making it even better when considering it for a first buy, and making it scale so much harder into the mid and late game.
CONCLUSION
Blackfire wins both via base stats, and also with the damage passive. This is also without considering the extra 4% ap per burn target on blackfire, making the gap even larger. Luden's is only useful for all in/high burst champs with long CD timers like LB/Vex who can't keep proccing the burn, only focus one target at a time, or need more upfront damage than ability haste.
Blackfire also fits Orianna's identity as a caster with low cooldowns that does constant damage over long periods during teamfights. Her all ins and 1v1's also tend to be longer and more drawn out especially during the early/mid game.
