I've played it both ways, but I've saved Kaidan more than Ash because not only do I like him better, his skill set is more valuable to me than Ashley's.
I'm not a huge fan of Ashley's personality. I respect the strong female soldier personality and her focus, but she just seems like one of those people I tend to have a lot of friction with. I'm more of a "think first, and act accordingly" type of person, and Ash is very much a "leap before you look" person. But that's just me, I know others feel differently, and that's cool too.
Now when it comes to who would be a better Spectre...
Let's say you have two employees. Employee A is fast and efficient at their job, but isn't so good at customer service and dealing with difficult people. Employee B is the best customer service rep on your team and they can handle difficult customers, but maybe they aren't as fast as your other employees (but they're still good at their job nonetheless).
They're both valued members of your team because they both excel in their own way and you can use them both in a variety of situations.
Now let's say you're looking to promote either Employee A or Employee B to be the manager of your company. The efficiency of Emloyee A is undeniable and you'd be lost without it, but you also know that people who are damn good at customer service are hard to find and Employee B represents your company better than anybody else on your team. Do you choose someone who can get the job done like no one else, or do you choose the person who paints a good picture of your company?
I think they both have qualities. I like the idea of Ashley becoming a Spectre and kind of getting redemption for her family that way. The Williams' were shamed for so long, and for a Williams to become the second human Spectre, that's kind of a big "eff you" to the system that looked down upon them. She's a career military woman and she will do what she has to do to get the job done. For me, story wise, I prefer leaving Ash on Virmire and getting the redemption for her family that way. But I think the Spectre route works too.
Kaidan is a little less hotheaded and more reasonable than Ash. On Mars and in the hospital, he doesn't jump on you like Ash does for working with Cerberus, and he's more willing to put that in the past. He's just a nice guy who is more willing to listen through until the end before making a conclusion, whereas Ash is more likely to get halfway through the story and then shoot.
I highly doubt all Spectres have the same methods. It's said by many that some Spectres act as if they are "above the law", and perhaps they are and the Council turns a blind eye to them because they get results and that's all they want. I think Ashley would get results faster than Kaidan, but maybe not in the cleanest or most orthodox of means. Kaidan would probably be respected for being a little gentler but also able to bring the heat when necessary.
I feel like Kaidan would be more likely to hesitate, whereas Ash won't take a chance and will plant a bullet between your eyes just to be safe.
Both would be fine Spectres. If you're playing Paragon, I think Kaidan would be the guy you'd want for the job. Renegade, Ashley maybe fits the bill more.