The Form 1041 instructions delineate the differences rather succinctly but one of the primary differences is that a complex trust can usually distribute capital gains to the beneficiary(ies), which otherwise belong to the corpus of the trust and remain within the trust.
See https://www.irs.gov/instructions/i1041#idm140376226642960
A couple of things to bear in mind include:
1) The trustee (or administrator) does not get to choose "simple" or "complex"; that is contained in the trust document.
2) In the final year of the trust, the trust is automatically complex (since it will be distributing corpus and terminating).
