The first difference is that Irish whiskey is generally distilled three times, making it smoother, though that's not required. Scotch is only distilled twice.
In Irish whiskey, modern producers don't generally let the peat smoke from the drying fire come in contact with the barley. This prevents Irish whiskey from developing a peat-smoke flavor.
On the other hand, with Scotch, the peat smoke is allowed to come in contact with the grain when helps give the spirit its trademark (but not required) smokiness.