If you refer $\int_{(0,\,1)} y \,dx$ to the Riemann integral of $y: (0,\,1) \to \mathbb{R}$ over $(0,\,1)$, there is no such a thing. Since Riemann integral is only defined on a closed interval. But if we refer the case to general integral, it works for Lebesgue integral, and it has the same value as $\int_{[0,\,1]} y \,dx$, as we will see below.

Let me invoke measure theory to answer the question. First, suppose $y: [0,\, 1] \to \mathbb{R}$ is Riemann integrable. In measure theory, every Riemann integrable function is a Lebesgue integrable function. And a function $f: X \to \mathbb{R}$ is Lebesgue integrable if and only if $$ \int_X|f| \,d\mu < \infty \,. $$ And of course, $f$ must be measurable (i.e., for every measurable set $A \subseteq \mathbb{R}$, $f^{-1}(A)$ is measurable on $X$). Hence $y$ is Lebesgue integrable by definition.

And note that every singleton in $\mathbb{R}$ has Lebesgue measure zero, i. e., $\mu(\{x\}) = 0$ for every $x \in \mathbb{R}$. Then we also have $[0,\, 1] = (0,\, 1) \cup \{0, 1\}$ and $(0,\, 1) \cap \{0, 1\} = \varnothing$, which means that $\{(0,\, 1),\, \{0, 1\}\}$ is a partition of $[0,\, 1]$.

Then by simple function approximation theorem, for every $k \in \mathbb{N}$, there exist measurable simple functions $s_k^+, s_k^-: [0,\, 1] \to [0,\, 1)$ such that $$ \forall x \in [0,\, 1]\,\big[ 0 \leq s_1^+(x) \leq s_2^+(x) \dotsb \leq y^+(x) \big] \;\land\; \forall x \in [0,\, 1] \left[ \lim_{k \to \infty} s_k^+(x) = y^+(x) \right] $$ and $$ \forall x \in [0,\, 1]\,\big[ 0 \leq s_1^-(x) \leq s_2^-(x) \dotsb \leq y^-(x) \big] \;\land\; \forall x \in [0,\, 1] \left[ \lim_{k \to \infty} s_k^-(x) = y^-(x) \right] $$ where $y^+ := \max \{y, 0\}$ and $y^- := \max \{-y, 0\}$. Note that $y^+$ and $y^-$ are measurable since $y$ is Lebesgue integrable. Then for every $k \in \mathbb{N}$ and $* \in \{+, -\}$, let $m_k^* \in \mathbb{N}$, and we define $$ \mathrm{image}(s_k^*) := \{\alpha_{(k, 1)}^*, \dotsc, \alpha_{(k, m_k)}^*\} $$ and $$ \forall j \in \{1, \dotsc, m_k^*\} \,,\; A_{(k,\, j)}^* := (s_k^*)^{-1}(\{\alpha_{(k,\, j)}^*\}) \,, $$ and then we can express $s_k^*$ by $$ s_k^* = \sum_{j = 1}^{m_k^*} \alpha_{(k, j)}^* \chi_{A_{(k, j)}^*} \,, $$ where $\chi$ is an indicator function. Note that $\left\{A_{(k, j)}^*\right\}_{j = 1}^{m_k^*}$ forms a partition of $[0,\,1]$, for every $k \in \mathbb{N}$ and $* \in \{+, -\}$. And since $A_{(k, j)}^* \cap \{0\} \subseteq \{0\}$ and $A_{(k, j)}^* \cap \{1\} \subseteq \{1\}$, hence we have $$ 0 \leq \mu(A_{(k, j)}^* \cap \{0\}) \leq \mu(\{0\}) = 0 $$ and $$ 0 \leq \mu(A_{(k, j)}^* \cap \{1\}) \leq \mu(\{1\}) = 0 \,, $$ which show that $\mu(A_{(k, j)}^* \cap \{0\}) = 0$ and $\mu(A_{(k, j)}^* \cap \{1\}) = 0$, for every $j \in \{1, \dotsc, m_k^*\}$.

Then by the property of Lebesgue integral we obtain \begin{align} \int_{[0,\, 1]} y\,d\mu &= \int_{\{0\} \cup (0,\, 1) \cup \{1\}} y \,d\mu \\ &= \int_{\{0\}} y\,d\mu + \int_{(0,\, 1)} y \,d\mu + \int_{\{1\}} y \,d\mu \\ &= \left[ \int_{\{0\}} y^+ \,d\mu - \int_{\{0\}} y^- \,d\mu \right] + \left[ \int_{(0,\,1)} y^+ \,d\mu - \int_{(0,\,1)} y^- \,d\mu \right] + \left[ \int_{\{1\}} y^+ \,d\mu - \int_{\{1\}} y^- \,d\mu \right] \\ &= \left[ \sup_{k \in \mathbb{N}} \int_{\{0\}} s_k^+ \,d\mu -\sup_{k \in \mathbb{N}} \int_{\{0\}} s_k^- \,d\mu \right] + \left[\sup_{k \in \mathbb{N}} \int_{(0,\,1)} s_k^+ \,d\mu - \sup_{k \in \mathbb{N}} \int_{(0,\,1)} s_k^- \,d\mu \right] \\&\quad\quad + \left[\sup_{k \in \mathbb{N}} \int_{\{1\}} s_k^+ \,d\mu - \sup_{k \in \mathbb{N}} \int_{\{1\}} s_k^- \,d\mu \right] \\ &= \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \mu\left(A_{(k, j)}^+ \cap \{0\}\right)\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \mu\left(A_{(k, j)}^- \cap \{0\}\right)\right) \right] \\ &\quad\quad+ \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \mu\left(A_{(k, j)}^+ \cap (0,\,1)\right)\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \mu\left(A_{(k, j)}^- \cap (0,\,1)\right)\right) \right] \\ &\quad\quad+ \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \mu\left(A_{(k, j)}^+ \cap \{1\}\right)\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \mu\left(A_{(k, j)}^- \cap \{1\}\right)\right) \right] \\ &= \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \cdot 0\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \cdot 0\right) \right] \\ &\quad\quad+ \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \mu\left(A_{(k, j)}^+ \cap (0,\,1)\right)\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \mu\left(A_{(k, j)}^- \cap (0,\,1)\right)\right) \right] \\ &\quad\quad+ \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \cdot 0\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \cdot 0\right) \right] \\ &= (0 - 0) + \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \mu\left(A_{(k, j)}^+ \cap (0,\,1)\right)\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \mu\left(A_{(k, j)}^- \cap (0,\,1)\right)\right) \right] \\ &\quad\quad +(0 - 0) \\ &= \left[ \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^+} \alpha_{(k, j)}^+ \mu\left(A_{(k, j)}^+ \cap (0,\,1)\right)\right) - \left(\sup_{k \in \mathbb{N}} \sum_{j = 1}^{m_k^-} \alpha_{(k, j)}^- \mu\left(A_{(k, j)}^- \cap (0,\,1)\right)\right) \right]\\ &= \sup_{k \in \mathbb{N}} \int_{(0,\,1)} s_k^+ \,d\mu - \sup_{k \in \mathbb{N}} \int_{(0,\,1)} s_k^- \,d\mu \\ &= \int_{(0,\,1)} y^+ \,d\mu - \int_{(0,\,1)} y^- \,d\mu \\ &= \int_{(0,\,1)} y \,d\mu \,. \end{align}

Recalling that $y$ is also Riemann integrable, then the Lebesgue integral is equal to the Riemann integral. And hence we obtain $$ \int_{(0,\,1)} y \,d\mu = \int_{[0,\,1]} y \,d\mu = \int_0^1 y \,dx \,. $$

I hope this answer could help. $\Box$

This is not a definitive answer, considering an answer have already accepted almost 2 years ago, but I believe it should be noted that in the case where there exists a Dirac delta function or any similarly defined special functions inside the integrand, the integral over some 'points' become non-zero by definition.

Using this fact, we can prove that the claim that these two integrals are equal leads to a contradiction if one or both of the limits of the interval are included a specially defined function. In this case, inclusion of a particular point will change the answer of the integration by a finite amount. Choosing this point from the limits of the interval, integration over an open interval will yield a different result than integration over an open interval by that finite amount.

So, integration on open or closed intervals can yield different results if there is a special function in the integrand such as Dirac delta.

Disclaimer: Notion of integrating δ(x) over an interval [0,a] might be undefined on its own. If anybody have any information in this regard, please edit the answer accordingly.

See also: Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

There is not difference. $$\int_{[0,1]}f(x).dx = \int_{(0,1)}f(x).dx = \lim_{n\to \infty} \left(I_n^+(f,0,{1\over 2}) + I_n^-(f,{1\over 2},1)\right) $$

With $I_n^+$ and $I_n^-$ respectively the upper and lower Riemann sums ( I chose such a decomposition because those Riemann sums never involve $f(0)$ and $f(1)$ )