Through and thru are different spellings of the same word. Thru is the less preferred form, however, and it might be considered out of place outside the most informal contexts. If you’re writing for school or for a job application, for instance, through is definitely the safer choice.
One exception: The shorter spelling is often used in drive-thru, where the term relates to getting fast food or banking without exiting one’s car. But though the shorter spelling has gained ground in this use, drive-through still prevails by a significant margin.
In current news publications that make content available online, thru only appears a tiny fraction of the time. We find only a few scattered instances of its use, against tens of thousands of instances of through. Thru is certainly gaining ground in text-speak and social networking, and it may someday become the preferred spelling, but we’re not betting on it just yet.
The ngram below graphs the use of through and thru in a large number of English-language texts published from 1800 to 2000. As you can see, thru barely registers against through (though it has a brief spike around 1920, which we can’t explain).