What happens if you don't bond neutral and ground in a main service panel?
Maybe if you owned the transformer

Imagine you had total control of the transformer.  (which you probably don't). You would be able to assure that the neutral-ground bond did not exist anywhere.  If you accomplished this, and didn't have any defects in your wiring, then you would have an isolated system which I discuss here. 

There are advantages and disadvantages. As I discuss, the first ground-fault isn't dangerous. It merely biases the system (like a ground strap would do); just at an unexpected and unplanned voltage.  A hot-ground fault pegs that hot as 0V from ground; neutral is 120V and the other hot is 240V.  On the other hand, a supply transformer primary-secondary fault pegs your wires at 2400V from ground.  Is your toaster insulated for 2400V?  I'm guessing not. 

Another disadvantage I didn't discuss is that -- remember that power wants to return to source, not ground.  Except lightning does want to get to ground.  So does ESD.  If your toaster isn't insulated for 2400V, it's probably not ready for a 50kA lightning hit.  Shuffle your feet across the carpet and zap the toaster chassis with static electricity, where does it go?  Can't go to ground, the toaster has a 2-prong cord. Without a neutral-ground bond, it will be efficiently carried to every device, even the ones that are "turned off" - neutral isn't switched. 

Remember, hot and neutral are not isolated -- they are bonded, with a bias.  A 120V bias assuming the transformer is turned on.  A transformer winding has very low resistance.

... Hazardous if you don't own the transformer

Most likely, the transformer is owned by the power company, and they have a neutral-ground bond at the transformer.  And several other houses are served by the same transformer, and each has a neutral-ground bond.   

So what happens if you have a bolted (zero ohm) hot-ground fault? 

It's flowing into your grounding system, as intended.  It wants to get back to source (neutral).  It will follow all available paths in inverse proportion of their resistance.  Where does it go?  Not directly to neutral, because you didn't bond it.  So where? 

"Okay, it'll go through my healthy grounding system, to Earth, across Earth to a neighbor or the transformer.  Because Earth is magic."  Nope -- earth is dirt. Dirt is a lousy conductor. 

Not enough current will flow to pull your grounding system back to 0V where it belongs.  So it will raise the voltage of your grounding system to near 120.

Now every grounded surface that was meant to be safe - the cover plate screws on light switches, your PC, your refrigerator, your dishwasher - is now energized near 120V.  Via the grounding rod, it also raises the voltage of the earth around your house to 120V, causing all kinds of weirdness. 

What's more, because it won't flow enough current (20A) to trip a breaker, it won't trip a breaker. So this fault will continue indefinitely.  How would you even know it was there?  You'd live normally until something extreme brought it to your notice.  What would that even be? 

I've heard of a situation where a landline phone wouldn't ring. The repairman, calling to confirm a repair date, asked how the customer knew to pick up the phone.  "My dog yelps", they said. 

Uncontained electricity is crazy stuff.