Early propagation of carries in the BLAKE hash function The BLAKE paper uses round constants derived from $\pi$ to improve diffusion, even if the additions modulo $2^{32}$ are replaced with bitwise XOR when processing a null message and null initial state:


  The higher weight in the original model is due to the addition carries induced by the constants $c_0,\dots,c_{15}$. A technique to avoid carries at the first round and get a low-weight output difference is to choose a message such that $m_0 = c_0,\dots,m_{15} = c_{15}$. At the subsequent rounds, however, nonzero words are introduced because of the different permutations.


However, the BLAKE2 paper describes changes it made to BLAKE that remove these constants:


  Constants in G initially aimed to guarantee early propagation of carries, but it turned out that the benefits (if any) are not worth the performance penalty.  This change saves two xors and two loads per G, that is, 16% of the total arithmetic (addition and xor) instructions.


This statement was made as part of an argument that reducing the number of constants in BLAKE2 does not weaken it, while simultaneously decreasing RAM and ROM footprint. Unfortunately, unlike many of the other trade-offs it discusses, no further explanation is given. What are the benefits of guaranteeing early propagation of carries that lead to BLAKE's constants, and why are these benefits in doubt?
This is already covered by Ella's response, but in short the added nonlinearity at the start buys you very little in terms of security, compared to its cost in number of operations. It would be more effective, and make for a simpler design, for example, to trade all these constant xors by an extra round. Since that part of BLAKE was not pulling its own weight, as noted by the cryptanalysis of the various weakened versions of BLAKE, it was removed.

But besides the small effect on nonlinearity, there is another reason for the existence of the constants—to eliminate symmetries on the underlying block cipher. BLAKE, as well as BLAKE2, are modes of operation on top of a $\{0,1\}^{1024}\times\{0,1\}^{1024}\to \{0,1\}^{1024}$-bit (or $\{0,1\}^{512}\times\{0,1\}^{512}\to\{0,1\}^{512}$) block cipher $E(k, m)$, inspired by the ChaCha core. And like the ChaCha core, when the message words $m_i$ are all 0, or more generally all equal, if the inputs $x_i$ are also, say, all equal, the output words $y_i$ will also be all equal. For example,
$$
E\left(
\begin{pmatrix}
m &amp; m &amp; m &amp; m \\
m &amp; m &amp; m &amp; m \\
m &amp; m &amp; m &amp; m \\
m &amp; m &amp; m &amp; m \\
\end{pmatrix},
\begin{pmatrix}
x &amp; x &amp; x &amp; x \\
x &amp; x &amp; x &amp; x \\
x &amp; x &amp; x &amp; x \\
x &amp; x &amp; x &amp; x \\
\end{pmatrix}
\right) =
\begin{pmatrix}
y &amp; y &amp; y &amp; y \\
y &amp; y &amp; y &amp; y \\
y &amp; y &amp; y &amp; y \\
y &amp; y &amp; y &amp; y \\
\end{pmatrix}\,,
$$

for some 32- or 64-bit word $m$, $x$, and $y$. This property, often called a symmetry of the round function, makes the underlying block cipher of BLAKE—without the constants—easily distinguished from an ideal cipher. Likewise with BLAKE2. This is even exploited in the "chosen-IV" attacks of a weak variant of BLAKE2. 

However, in the actual mode of operation of either BLAKE or BLAKE2, these so-called symmetric states are unreachable by design, via the inclusion of the $IV_0$–$IV_7$ constants in the initial state, so this property is not a security liability.