Of course the 2nd one doesn't find a match, there's no status: text in the value. I think you wanted:
select * from students where data_json LIKE '%"status":%'
... however, like most cases where you attempt text pattern matching on structured data this is in general a terrible idea that will bite you. Just a couple of problem examples:
{
"somekey": "the value is \"status\": true"
}
... where "status": appears as part of the text value and will match even though it shouldn't, and:
{
status : "blah"
}
where status has no quotes and a space between the quotes and colon. As far as JavaScript is concerned this is the same as "status": but it won't match.
If you're trying to find fields within json or extract fields from json, do it with a json parser. PL/V8 may be of interest, or the json libraries available for tools like pl/perl, pl/pythonu, etc. Future PostgreSQL versions will have functions to get a json key by path, test if a json value exists, etc, but 9.2 does not.
At this point you might be thinking "why don't I use regular expressions". Don't go there, you do not want to try to write a full JSON parser in regex. this blog entry is somewhat relevant.