This became more detailed and elaborate then I originally set out for it to be. For those who can't be bothered to read more than a few sentences, the definition of a blowout is P-2, the definition of a burst is P-3, the similarities between the two are in P-1, a dissertation on why you are all endlessly confusing one another is P-5 and if you want the conclusion to that and not the history lesson, skip paragraphs 6 & 7 and go straight to 8. Following P-8 is a list of tire failures, elaborated upon as simply as possible using basic equalities/inequalities and amusing analogies. If you can't figure it out after that, just stop trying.
In my opinion, (The engineer/classic muscle car purist opinion):
P-1
Both are slang for a tire catastrophically failing. Not to be confused with a punctured or flat tire (easily repaired), they're a significant rupture that occurs in the tread or sidewall in a very short amount of time (think instantaneous) causing a complete loss of air pressure. If the vehicle is in motion during such an event, the tire subsequently shreds to pieces due to excessive friction with the contact surface. Here's where the difference is.
P-2
A Blowout can be caused by a number of factors: brittle or cracked rubber, contact with a foreign object at speed, excessive heat due to low tire pressure, excessive wear (which can include a neglected or poorly repaired puncture) or excessive stress (drifting/e-brake turns). When one or more of these exceeds the strength of the tire, weakens it to the point that it cannot endure typical operation or a combination of those two, the tire fails and BLOWS OUT all of the air. It does not necessarily have to be loud and oftentimes isn't.
P-3
A Burst is always caused by excessive air pressure. This is caused by the tire being greatly overinflated, a deformation of the tire itself (think squeezing a balloon) or a combination of the two. The pressure literally tears the metal cord and rubber apart always at the weakest point. The most common causes would be potholes (while it only falls an inch or two, a 1 ton assembly tends to squeeze things a bit) or the primary operator being a dumb ____ and not reading the pressure rating printed on the sidewall. It could also happen during a drift (the tire deforms along its width not diameter) though I cannot say I've heard of such a case personally. The most distinguishing difference is that a burst is ALWAYS loud. The tire has literally exploded, and the greater the pressure the louder the bang.
P-4
I remember I was fixing up an old bike with beat up 90psi tires; one burst at 60psi and it was louder than a launch charge on a 500 gram mortar (very big firework). My ears were ringing for a full hour after.
P-5
Here's the part that's confusing you: When you further condense the slang term "blowout" to "blow", it no longer retains its original meaning. A blown tire could have either blown UP or blown OUT.
P-6
The first phrase has been around for over a century at the very least and has been historically been used to describe an explosion (though large parties, landslide victories and vocal outbursts are also appropriate uses for it). While you wouldn't traditionally say that your tire blew up, the phrase is a completely accurate and acceptable summary of the event. It is synonymous with "burst".
P-7
"Blowout" is a much younger term and while it's recognized by even the most conservative English dictionaries, at some point it was just a completely arbitrary junction of words someone felt accurately described tire failure. The popularity and widespread use of that slang eventually made it a 'real' word.
P-8
For this reason, we have two compound phrases with the same leading verb (blow up & blow out) used to describe two practically opposite instances of tire failure (one caused by outside forces tearing in, the other by inside forces tearing out). Words can have different meanings with different context, it is your job when interpreting a sentence to apply the meaning that fits. As a result, the word blown in the context of a tire destroyed from use is always a true statement. Without the specification of "up" or "out", it means both phrases simultaneously until given the context to interpret it with.
A recap (1-3), clarification (4-5) and expansion (5+) in terms of water balloons:
blow ≈ blowout -----------> Ex: Guy has a water balloon and untied the knot.
blow ≈ blow-up = burst --> Ex: Guy has a water balloon, put a grenade inside and pulled the pin.
burst ≠ blowout ------------> Why? In one of those examples Guy doesn't have hands.
burst ≈ blowout ------------> In both examples, the balloon won't hold water.
burst - blowout ≈ 0 --------> Difference? None, unless your name is Guy.
puncture -------------------> Guy poked a pin through the water balloon. It leaks now.
puncture + exaggeration> The story Guy tells his friends is "I blew a hole in my water balloon."
--------------------------------> Guy uses ridiculous phrases like "brah" and "right on" during the retelling
--------------------------------> to compensate for a lack of masculinity and sense of self worth.
puncture + time = flat ----> The water balloon is noticeably less than full of water.
flat + flat = dismounted -> Guy neglected the water level so vigilantly that it fell off its proverbial rim.
--------------------------------> About as much effort went into making that balloon related as you would --------------------------------> need to prevent such a event from occurring in the first place.
puncture + patch = 0 ----> Guy filled the hole of one old, rubber balloon...
--------------------------------...with the material of a second rubber balloon...
--------------------------------...then covered it with liquid balloon on both sides...
--------------------------------...then smothered it all with rubber cement...
--------------------------------...before wondering if it's time to buy a new balloon. (Still, as good as before.)
patch - fix = fix-a-flat ----> Guy heard that if he filled his balloon with this thick, sticky goo it won't leak.
--------------------------------> Doesn't sound like a sketchy, half-assed fix to me! (Not even a real solution.)
use + friction = worn -----> The water balloon has become smoother over time.
-------------------------------->...because it wasn't to start with... A responsible man would replace it.
use + worn = bald --------> The possibility of failure doesn't deter a true cheapskate such as Guy.
--------------------------------> The water balloon is unquestionably smoother than a baby's ass.
use + bald = raw braid --> Huh, that's funny. The water balloon makes a crunching sound when rolled.
--------------------------------> Is that the smell of exposed steel or impending doom?
time + time = cracking --> Guy left his water balloon outside untouched for the better part of a decade.
--------------------------------> To his surprise, the weak and unpatchable areas left him death threats.
-------------------------------->"When you least expect it..." it read. At least it was on a Hallmark card.
Valve stem failure -------> Guy has been through some pretty tough times with that water balloon.
--------------------------------> Thought he'd seen everything... and then it started leaking through the knot.
--------------------------------> "How, wha....whe..why is this even possible!?"
--------------------------------> Because ____ you Guy, that's why.
