The meanings are different.
This looks as though it is the command description for a remove function for a list class, so I'll explain with reference to this.
The word occurrence can be read as instance or, informally, 'time you see it'.
In the context of a remove function, this could be explained simply: beginning from the start of the list, the program looks at each object in the list until it finds [an occurrence of] the object you want to remove. It removes [this occurrence of] this object and then stops looking. There might or might not be more [occurrences] of the object you want want to remove in the list afterwards - the program doesn't check this.
Including the bracketed phrases makes the explanation slightly more complicated but more accurate.
If you instead explained the function as 'Removes the first encountered object from list' then there is no 'specific object'. In programming terms, the function wouldn't need to take an argument. This would mean: beginning from the start of the list, remove the first object from the list (depending on how the language handles nulls, this may not be the first element of the list).
'Removes the first encountered object that is equal to a specific object from list' would have essentially the same meaning (assuming the language defines equality as having the same reference), but the original version is clearer and shorter.