Almost all Tristana players I've seen rush IE. Almost all Vayne players I've seen rush BotRK.
Both are champions that are almost exclusively Auto Attack based.
Is there any reason for for this difference?
Vayne scales well with attack speed, due to her true damage. Instead of building for large crits you build for rapid application of true damage.
I'm struggling to think of a good metaphor for this situation, but imagine you're playing a resource game where you can buy things with gems, gold, or iron, but you can only buy advantages with one or two of those, not all three. If you spread yourself across all three, you'll have lots of resources but you won't be able to buy the expensive things in any one category.
That's kind of like how damage is in League. True, physical, magical, attack speed, flat damage, crit damage. Spreading yourself across all of them, you encounter the resistances of all of them (except true, of course), and the benefits and downfalls of each.
Vayne is not a crit monster, not really. Her goal is not to crit (she will after 2 PD and an IE), it's to proc her silver bolts. Late in the game she crits, but she's not going to be critting for the kind of damage that Trist does because they build differently. Vayne is about true damage from silver bolts (percent health) and percent health from Bork, both of which cannot be increased with AD but CAN be increased with attack speed. So you focus on doing percent damage with her.
Trist has no innate percent damage, and she has attack speed in her kit, so she wants to focus more on filling in the "missing" damage (compared to vayne, who gets damage for free instead of AS). The way she does that is by acquiring lots of AD and crit, some armor pen, and a little AS. So in other words, Vayne has damage, she just needs a little more, and lots of AS. Trist has AS, needs a little more, and lots of AD.
Getting a Bork on Trist isn't bad per se. If you get it early enough (like first item after you get early kills) you can leverage the relatively large damage boost (80 flat damage per 1000 hp) into a lead. It will eventually fall off though (even, IMO, against tanky opponents). So like you want a bork before they finish their BF sword. Otherwise, that extra damage from the bork does nothing for Trist's crit damage, which is really what she is there to do. Late game, high speed crits.
Tristana doesn't need the copious AS boost due to her Q and Stakikk Shiv/Phantom Dancer being more than enough. The active is also too short-ranged and irrelevant since her W and R are also massively powerful disengage tools with superior effectiveness.
Vayne, on the other hand, has nothing beyond 550 range, and her Q and E aren't as powerful disengage. On top of that, she needs a ton of AS, and also her W synergizes very well with the %health damage of BotRK.
There's a lot of reasons but there are two big ones:
It's funny that everyone talks about the true damage, but in most of her games Vayne's overall damage is only about 20-22% true damage. People build for it, but it only accounts for about 1/5 of your total damage output.
Probably because Trist has her own AS boost and can focus on AD. Sometimes ppl get Shiv on her too which would make BotRK even less needed. At least that's my theory.
Trist outranges the active.
Vayne is the supreme anti-tank ADC, and coincidentally, Blade of the Ruined King is an anti tank item who's anti tank abilities scale on the same stats as Vayne's (in this case, Attack Speed).
Trist owns attack speed boost, so building AD and crit will help her early, + his pushing ability due explosive shot. In the other hand silver bolts true damage + the high ad scaling Vayne has makes BOTRK a perfect buy, adding life steal, attack speed and active for dueling to its arsenal.
Vayne scales off AS Hence why you see. 
But Tristana has a huge AS buff on top of safe damage. So you build her for crit damage early and then the comes in when they have alot ofhp and you need to tank melt. If they have no HP you can go ahead with 
Not to mention she benefits immensely from the active to chase down opponents. Trist just jumps at em.
I find Trist doesn't use the active as much as other closer range can and doesn't require the very important slow. Her goal is to W away and then ult away while maintaining dps. She certainly can use the active but why does she need it?
Vayne has true damage, she scales well with attack speed and lifesteal. Tristana doesn't have true damage, so crit and AD really helps her increase her damage. Her Q already gives her a ton of attack speed too. Also, she has a lot of range, late game, it wouldn't make sense for Tristana to move closer just to use BOTRK.
