I'm not a specialist in the field, but recently it happened to me to read the beautiful paper by Ono "The last words of a genius" on the Notices of the AMS, December 2010, which seems 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 form, but it is still a very interesting object. For instance, when $k=1/2$ it is a so-called *mock theta function*.

Mock theta functions were first described by Ramanujan in a famous letter to Hardy, written on his deathbed, but only very recently their deep connections with real-analytic modular forms were discovered by S. Zwegers, in his Ph.D. thesis written under D. Zagier.

For further detail you can look at Ono's paper or at the article "What is... a mock modular form?" by Amanda Folsom in the same issue of the Notices of the AMS.