It seems to me that the second Chebyshev function $\psi(x) = \sum_{n \le x} \Lambda(n)$ (where $\Lambda$ is the von Mangoldt function) is more natural, but as Gjergji says the two approximate each other. This is because the behavior of the second Chebyshev function is related by certain general theorems that I'm not familiar with to the behavior of the Dirichlet series

$$\sum_{n \ge 1} \frac{\Lambda(n)}{n^s}$$

and this Dirichlet series is precisely $\frac{-\zeta'(s)}{\zeta(s)}$, or the negative logarithmic derivative of $\zeta(s)$. This has the following intuitive interpretation: if we think of $\zeta(s)$ as the partition function

$$\zeta(s) = \sum_{n \ge 1} e^{-s \log n}$$

of the Riemann gas, then the negative logarithmic derivative of a partition function describes the expected value of energy at a given temperature, a fundamental property.

Regarding why it is natural to assign the state $n$ the energy $\log n$, the point is that for $s > 1$ we get the only probability distributions on the natural numbers which satisfy certain natural properties; see this blog post, for example.

Of course a basic purely mathematical reason to consider the logarithmic derivative is the ideas around the argument principle.

It seems to me that the second Chebyshev function $\psi(x) = \sum_{n \le x} \Lambda(n)$ (where $\Lambda$ is the von Mangoldt function) is more natural, but as Gjergji says the two approximate each other. This is because the behavior of the second Chebyshev function is related by certain general theorems that I'm not familiar with to the behavior of the Dirichlet series

$$\sum_{n \ge 1} \frac{\Lambda(n)}{n^s}$$

and this Dirichlet series is precisely $\frac{-\zeta'(s)}{\zeta(s)}$, or the negative logarithmic derivative of $\zeta(s)$. This has the following intuitive interpretation: if we think of $\zeta(s)$ as the partition function

$$\zeta(s) = \sum_{n \ge 1} e^{-s \log n}$$

of the Riemann gas, then the negative logarithmic derivative of a partition function describes the expected value of energy at a given temperature, a fundamental property.

Edit: This observation goes at least as far back as Mackey, Unitary Group Representations in Physics, Probability, and Number Theory (1978), who wrote:

...Our main point here is that one could have been led to the main outline of the proof of the prime number theorem by using the physical interpretation of Laplace transforms provided by statistical mechanics. In particular, the function $-\frac{\zeta'}{\zeta}$ whose representation as a Dirichlet series (Laplace transform with discrete measure) plays a central role in the proof has a direct physical interpretation as the internal energy function.

Regarding why it is natural to assign the state $n$ the energy $\log n$, the point is that for $s > 1$ we get the only probability distributions on the natural numbers which satisfy certain natural properties; see this blog post, for example.

Of course a basic purely mathematical reason to consider the logarithmic derivative is the ideas around the argument principle.

It seems to me that the second Chebyshev function $\psi(x) = \sum_{n \le x} \Lambda(n)$ (where $\Lambda$ is the von Mangoldt function) is more natural, but as Gjergji says the two approximate each other. This is because the behavior of the second Chebyshev function is related by certain general theorems that I'm not familiar with to the behavior of the Dirichlet series

$$\sum_{n \ge 1} \frac{\Lambda(n)}{n^s}$$

and this Dirichlet series is precisely $\frac{-\zeta'(s)}{\zeta(s)}$, or the negative logarithmic derivative of $\zeta(s)$. This has the following intuitive interpretation: if we think of $\zeta(s)$ as the partition function

$$\zeta(s) = \sum_{n \ge 1} e^{-s \log n}$$

of the Riemann gas, then the negative logarithmic derivative of a partition function describes the expected value of energy at a given temperature, a fundamental property.

Regarding why it is natural to assign the state $n$ the energy $\log n$, the point is that for $s > 1$ we get the only probability distributions on the natural numbers which satisfy certain natural properties; see this blog post, for example.

It seems to me that the second Chebyshev function $\psi(x) = \sum_{n \le x} \Lambda(n)$ (where $\Lambda$ is the von Mangoldt function) is more natural, but as Gjergji says the two approximate each other. This is because the behavior of the second Chebyshev function is related by certain general theorems that I'm not familiar with to the behavior of the Dirichlet series

$$\sum_{n \ge 1} \frac{\Lambda(n)}{n^s}$$

and this Dirichlet series is precisely $\frac{-\zeta'(s)}{\zeta(s)}$, or the negative logarithmic derivative of $\zeta(s)$. This has the following intuitive interpretation: if we think of $\zeta(s)$ as the partition function

$$\zeta(s) = \sum_{n \ge 1} e^{-s \log n}$$

of the Riemann gas, then the negative logarithmic derivative of a partition function describes the expected value of energy at a given temperature, a fundamental property.

Regarding why it is natural to assign the state $n$ the energy $\log n$, the point is that for $s > 1$ we get the only probability distributions on the natural numbers which satisfy certain natural properties; see this blog post, for example.

Of course a basic purely mathematical reason to consider the logarithmic derivative is the ideas around the argument principle.