I'm not an expert, but recently I read the beautiful paper by Ono "The last words of a genius" on the Notices of the AMS, Dcember 2010, which is related to your question.
Let $M \colon \mathbb{H} \to \mathbb{C}$ be a smooth function that transforms like a weight $k$ modular form and such that $\Delta_k(M)=0$. Then we say that $M$ is a weight $k$ harmonic Maass form.
Any harmonic Maass form can be uniquely written as
$M=M^{+} + M^{-}$,
where $M^+$ is the holomorphic part and $M^-$ is the non-holomorphic part. Then the modular forms are exactly those harmonic Maass forms such that $M^-=0$.
In the general case, the holomorphic part of a harmonic Maass form is not a modular forms, but it is still a very interesting object. For instance, when $k=1/2$ it is a so-called mock theta function.