Full is an adjective, and means ‘containing a lot’: The theatre is completely full this evening. The room was full of books. Fill is a verb, and means ‘make or become full’.

What is the difference between filled and filled?

As nouns the difference between filling and filing is that filling is anything that is used to fill something while filing is any particle that has been removed by a file or similar implement; a shaving.

Is full up grammatically correct?

In short, “full up” has been around for quite a while, but the OED still considers it colloquial. We think it’s OK in speech and casual writing, but we’d go with “full” on formal occasions.