What is the difference between real and complex manifolds?

Holomorphic functions are much more "rigid" than smooth functions. For example, let $U \subset \mathbb{C}^n$ be a a connected open set and let $V_1,V_2 \subset U$ be open subsets whose closures are disjoint (think disjoint small balls). If $f : U \rightarrow \mathbb{R}$ is just a smooth function, then the values of $f$ on $V_1$ and $V_2$ have no relationship whatsoever; indeed, if $f_i : V_i \rightarrow \mathbb{R}$ is smooth for $i=1,2$, then you can find a smooth function on $U$ which restricts to $f_i$ on $V_i$. However, if $f : U \rightarrow \mathbb{C}$ is holomorphic, then there is a very strong relationship between the values of $f$ on $V_1$ and $V_2$. In fact, just knowing what $f$ does on $V_1$ determines what it does everywhere on $U$, no matter how small $V_1$ is!

For manifolds, this rigidity manifests itself in the fact that complex manifolds have "extra structure" that a naked smooth manifold does not have. As an important example of this, consider a $1$-dimensional complex manifold $X$ (thus $X$ is a 2d-dimensional real manifold, i.e. a topological surface). A naked smooth topological surface has no "geometry"; in particular, you can't measure lengths of tangent vectors on it or angles between tangent vectors. On a $1$-dimensional complex manifold $X$, there is still no notion of "length". However, amazingly there is a notion of "angle" between tangent vectors! In the end, this comes down to the fact that bijective holomorphic maps between open subsets of $\mathbb{C}$ actually are conformal, i.e. they preserve angles between tangent vectors. This implies that the usual conformal (=angle measurement) structure on $\mathbb{C}$ induces a conformal structure on $X$.

There's a sort of "standard" answer to this question, involving the notion of an almost complex manifold.

A complex manifold has the property that each tangent space is endowed with the structure of a complex vector space. Roughly speaking, the difference between a real vector space of dimension $2n$ and a complex vector space of dimension $n$ is that in the complex vector space, you know what it means to multiply by $i$. More formally, you can put a complex structure on any real vector space by choosing any linear transformation $J$ such that $J^2 = -1$, with $J$ serving as multiplication by $i$.

Note that this $J$ adds a lot of structure to a real vector space. For example, each vector $v$ now has an associated $2$-dimensional subspace, namely the span of $v$ and $Jv$ (i.e. the set of complex multiples of $v$), and it's possible to measure angles between vectors on this subspace. Also, we can now classify linear transformations of the vector space into two types -- those that commute with $J$ (i.e. complex linear transformations) and those that don't.

A complex manifold of dimension $n$ can be thought of as a real manifold of dimension $2n$ together with a smoothly varying choice of $J$ for each tangent space. Indeed, a real manifold endowed with such a smooth choice of $J$'s is called an almost complex manifold.

The word "almost" is pertinent because not every smooth choice of $J$'s actually yields a complex manifold. You have to choose a set of $J$'s that are "integrable" in the sense that a certain tensor defined in terms of $J$ vanishes. My impression is that the geometric difference between almost complex manifolds and complex manifolds is not well-understood.

«why holomorphicity is a stronger condition than smoothness?»

Becase there are fewer holomorphic functions than smooth ones.

If we were to consider atlases whose transition functions were polynomial, the restriction would be stronger. If we were to consider only affine linear transition functions..., well, you get the picture :D

Given any continuous function $f : M \to \mathbb R$ it can be approximated by a $C^\infty$-smooth function. But continuous functions $f : M \to \mathbb C$ can't be approximated by complex differentiable functions, since those are constant (provided $M$ is compact and connected).

That's the kind of thing that's different with complex manifolds. Complex differentiability is a huge restriction on functions, while real differentiability (at least from the perspective of topology) is not a major restriction.

Regarding existence of smooth structures, see:

https://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere

and

https://mathoverflow.net/questions/11664/complex-structure-on-sn

for examples of how existence of complex structures is a far more subtle question than that of smooth structures for "everyday" manifolds.