They can be interpreted as meaning rather different things. I have highlighted in bold below the quantifiers for the noun stampede:
The latest major stampede occurred in 2006 killing at least 346 and injuring 289 people.
The latest major stampede killing at least 346 and injuring 289 people occurred in 2006.
To use a database analogy, the first sentence gets the list of stampedes, filters for the ones which could be called 'major', chooses the most recent one, and then returns the information about it: it was in 2006, killed 346, and injured 289.
The second sentence gets the list of stampedes, filters for the ones which could be called 'major', but then also filters for those in which 346 people died and 289 were injured. It returns just one piece of information, the year. This would be a very odd thing to say, so it can also be interpreted as meaning essentially they same as the first sentence, with the causalities details being dislocated but in that case punctuation should be inserted to mark where pauses would be spoken:
The latest major stampede, killing at least 346 and injuring 289 people, occurred in 2006.
The latest major stampede (killing at least 346 and injuring 289 people) occurred in 2006.