I seriously love these kind of debates. They make life fun.
Before I go anywhere, I want to say two things: One, I will try my best to limit my bias, and bash both 'Frames equally. Two, I have never played either 'Frame before, which may aid me in eradicating my bias as much as possible.
First things first, let's address the main topic here: They're two different 'Frames. Stop comparing them so hard because they can both go invisible. If that was the case, why couldn't you just go ahead and do a LOKI vs. ASH vs. SHADE debate? (At least SHADE would heal your shields. Ever see a LOKI or ASH do that, huh? I thought so.)
Bam. That is the easiest summary I can make, and those who want the easy way out, the door is to your left, have a nice trip.
Now, let us continue on the perilous, rage-filled fahnboi/fahngurl biased nether-regions of the topic, the place we shouldn't touch, but we do anyway. I'm not sure if people still debate this, but people used to debate the "Nasty-Tasting Burnt Matter" and "Mischievous Low-Key" topic, and I wanted to create a place to do that (probably not the first time).
Now, we already know the basics of the two friendly neighborhood space ninjas, and the line should have already been drawn there:
LOKI is used for crowd-control and manipulation, using his cheap-arse ways of screwing with the enemies minds, rather than using directly damaging and painful methods of power, including the erecting of pretty affinity-patterned statues of himself, tripping allies and enemies with confusing teleports, and more, with his mighty skill-set of:
"Totally-Real-And-Not-Obvious" Decoy,
"Blandly-Named" Invisibility,
"Trolling-And-Disorientating-Diabolical-Trade-Places-With-Me-I'm-Too-Lazy-To-Run-To-Extraction" Teleport,
and the very underestimated and underappreciated...
"I-Stole-All-Your-Guns-What-Now-Oh-God-Where-Did-You-Get-Those-Batons" Disarm.
Of course, his signature is Invisibility (which is why we have this debate to begin with), which is everyone's favorite LOKI skill, and should be self-explanatory (if you need it explained, Invisibility allows LOKI to fly around by farting clouds of pure space-ninja rainbows out of his hammerhead submarine face, reaching heights of over 50 meters.)
ASH is stealth AND assault-based, so he got more than just "I faq wit jeu, ha no mur gunz, I go nao, bai *poof*", but is rather suited for "I fawk wert hew, I terlepoorpt errywhar, stabby yo fiac, I go nao, bai *poof*". ASH has the ability to super-ninja skills teleport unpleasantly into the faces of unsuspecting gooks, go invisible in a totally inconspicuous cloud of flour (Granny was not pleased, little ASH), and other stuff, with his mighty skill-set of:
"Totally-Clichéd-Ninja-Move-Spinny-Death" Shuriken,
"Granny's-Stolen-Cookie-Flour-Oh-God-It's-In-My-Pants" Bomb (Nobody actually wears pants in WARFRAME),
"I'm-In-Your-Bubble-Are-You-Uncomfortable-Please-Stop-Butting-My-Face-With-That-Gun-It-Hurts" Teleport,
and the very finicky, cinematic, cool, over-exaggerated and very slow-the-pod's-probably-dying...
"Dude-I'm-Creed-Style-On-This-See-My-Shiny-Wrist-Blades-Here-Have-A-Closer-Look-Whoops-You-Died-Got-To-Show-Someone-Else-Whoops-You-Died-Too" Storm.
ASH is a little less focused around his invisibility, but it is one of his biggest highlights, making the other half of this debate. All that wasted flour stirred up some arguments, and not just with Granny.
Now, if you really have curiosity of the direct info of these skills, have a poke at the conveniently linked wiki pages for both LOKI and ASH here.
Comparing the two, we can see an obvious break in skill-set here. Again, LOKI is that guy who dumps dirt in your nonexistent pants and runs, while ASH grinds your face into the said dirt. So, let's put a very basic and generic list of relative advantages/disadvantages.
I'm going to organize this guy like I organized my (outdated) LATRON vs. BOLTOR argument, with the "main topic" by a bullet (-) and following specifics by an asterisk (*). Less civilized names can be referred to as dashy-things and f**king stars.
LOKI:
-Best suited as a supportive 'Frame to aid and complement other 'Frames.
-Core abilities revolve around indirect damage, suppression, and manipulation. (Stealing change and shoplifting included.)
*Decoy and Switch teleport allow for manipulation, misdirecting and disorienting enemies. (Huh, that glowing statue looks totally real, let's shoot it.)
*Invisibility gives LOKI very handy escape, mobility, stealth, and indirect damage with the melee damage modifier. (Oh god, I'm getting cut to pieces by something I can't see, let's not be alerted by it, it'll pass. Probably just a breeze.)
*Radial Disarm suppresses the crowd, robbing them of their weapons, but giving them BATONS (Zap zap, LOKI, zap zap.), forcing them to move in close with their dumb-as-bricks AI, which allows the Ten-Gnomes to strike them down with great vengeance and furious anger to those who would attempt to poison and destroy their brothers (and sisters) with batons. {Totally not Ezekial 25:17, and totally not made up by Tarantino.}
-Faster than ASH. Heck, fastest frame in the game, except for that one on crack (also known as VOLT).
*Grants LOKI the ability to run in circles and run like a wimp to extraction. Higher mobility, higher survivability (debatably).
-Lower Health and Shields pool, with Lowest possible Health value in the game (Nekros won in Shields).
*Paper Mache armor technology allows literally no tanking whatsoever. Down in seconds. Keep that invisibility up.
-Energy pool of 150 base.
*Being totally skill based, this goes a long way. Adding Flow is still requirement, but more energy = more fun.
*The is perfect for a Redirection/Vitality, in case you need to remove a bit of your absurd squish-factor. If maximizing on duration, use Narrow Minded in this ideally fit polarity. The is good for a Continuity mod to further your power duration.
*Ideal for Energy Siphon. Go figure.
ASH:
-Better suited for a main damage 'Frame with some mobility.
-Core abilities revolve around direct and indirect damage and mobility. (Stabby stabby stab, then flour the floor.)
*Shuriken and Blade Storm both do direct damage, giving ASH the ability to attack enemies with tracking spinny-blades-of-death, and two if you max it out (which you should always max skills out). Blade Storm may be clunky and slow, and the pod/hostage will most likely die in the time allotted to dramatically and cinematically eradicate your opponents with AssassCreed Style, but the damage is pretty heavy for what it's worth, so if you have the time, or just need a pseudo-escape of invulnerability, it helps. (Oh god my friend got stabbed to death, and bullet's don't kill the guy. Let's shoot some more until my face gets stabbed in.)
*Smoke Bomb and Teleport can be used for mobility, with Flour Sack having the bonus of melee damage modifier, and staggering the enemies with the pure abject horror of flour flying everywhere. Teleport is a finicky skill, and most people forego it for another mod slot, but it can be used to get into people's faces (and get smacked across the face shortly thereafter).
-Fairly fast, but only the 3rd (no CrackFrame) fastest 'Frame, beaten by LOKI and NOVA.
*Still mobile enough to make it comfortable.
-Highest Health value, decent shields pool.
*Armor may be average, but the extra health makes great work for Vitality, allowing a fast and decently tanky 'Frame. Do not engage Toxic Ancients, for the last time.
-Energy pool of 100 base.
*The lesser of Energy pools, so slightly limited in terms of cast-spamming ability. Adding Flow will always help, as usual.
*Ideal for Focus, Continuity, Blind Rage, or Rage, depending on the build you aim for.
*Not as useful to most people, feel free to Rifle Amp or Steel Charge, or just Forma the slot to your preferred slot.
Even the polarities agree on the roles, for a good part, so choose depending on if you want to steal wallets or mug wallets. All up to you. 'Frames are not nearly as versatile as weapons, which can be modded out to convinience, giving you a different 180 role on some, but a 'Frame cannot alter its skill-set. This keeps LOKI and ASH apart, with the roles of support, versus the roles of assault.
These are my 2.5c on the topic, feel free to flak me down and burn my body, but keep it to a healthy and child-friendly argument, I guess.