(I started writing this post a few months ago but wanted to fix some details before I posted it, so the attack speed buff that she received in this patch had not yet been implemented when I wrote most of this.)
Innate: For each spell that's on cooldown, Ashe gains an additional 1 gold per lasthit.
Q: For the next four seconds, each of Ashe's basic attacks will slow her target for 2 seconds. Mana cost: 15. Cooldown: 6 / 5.5 / 5 / 4.5 / 4. Slow: 15% / 20% / 25% / 30% / 35%.
W: [No change]
E: Ashe animates a hawk to scout for her, revealing terrain for 2 seconds
as it flies towards a target location. Hawkshot reveals a wide area for 5
seconds when it reaches its destination. The hawk will reveal units in
the brushes, but will not reveal stealthed units or objects. In addition, Ashe's next 1/1/2/2/3 basic attacks will gain 100% crit chance. Range: 2500 / 3250 / 4000 / 4750 / 5500. Cooldown: 60.
R: [No change]
What have we done here? Basically, we've switched her innate and her E, and we've altered her Q so that it fits in with her new passive better. We've also increased the number of critical attacks that Ashe will have guaranteed for her in late-game teamfights as she levels her E.
What motivated these changes? Currently, Ashe has decent damage immediately upon entering lane but falls off after that, and so if you play slightly passively before she is forced to consume her first critical strike by last-hitting a creep, you can start to zone her very effectively due to the long initial cooldown of her W. Additionally, most of the massive threats to ADCs currently like Vi and Xin Zhao aren't countered by Ashe's slows enough to make her an appealing choice just for her slow late game.
These changes give her a slight increase in dps in her late-game damage by guaranteeing her 3 critical attacks at the beginning of each fight, but, more importantly, they allow her to decide when to use rare guaranteed crits in the early game. On the other hand, maxing E first will cost her 12 seconds of her W cooldown, and when she has about 75-100 AD, the 2-3 guaranteed crits only increases her overall burst damage by 75-100 or by 150-200, which is not enough to be an overpowered amount of burst early on. Overall, leveling E first will simply cost her too much DPS due to not having a decreased W cooldown.
Her new innate will give her slightly more gold than the passive on her E used to give, and she will be rewarded for using her cooldowns in lane. The gold increase will basically let her build a red elixir in lane.
Level 1-3 (Slightly less than 6 waves, or 6*6+2-3=35 CS): About 1 spell on cooldown. This is 35 more gold than you would currently get in levels 1-3
Level 4-5 (Slightly more than 6 waves, or 6*6+2+3=41 CS): About 2 spells on cooldown. This is 41 more gold than you would currently get in levels 4-5
Level 6+: About 2.5 spells on cooldown, or 1.5 extra gold per lasthit over what you currently get. To afford a red elixir, you need 250 gold, or an additional 116 CS worth of bonus gold over what Ashe currently earns.
So by the time laning is over, she should have made about a red elixir's worth of gold over what she would currently make during laning, maybe being able to buy an extra 1-2 potions also depending on how good her CS is. This red elixir will enable her to compete much more strongly with other ADCs who have better damages in early laning, or she can choose to simply be weaker for early laning and be slightly further ahead in her build.
Her Q now costs less mana if she is able to get 3 or 4 autoattacks off but more if she only gets 1. This makes it slightly more difficult for her to get a slow off on a single autoattack but allows her to spend less mana chasing someone down. This change was partially motivated just to fit in with her new passive, but it also makes her choose when to use her slow instead of just using it every single time she's able to attack an enemy champion, so she can be more effective as long as she plans accordingly but is now forced to decide how to use her skills more than she currently is.
Overall, this is a noticeable buff to Ashe, but as she's almost never seen in competitive play, I think it's appropriate.