Let's take the combinatorial definitions of delta set and simplicial set, as follows:
A delta set consists of a sequence of sets $X_0,X_1 \cdots$ and for each $n \geq 0$, functions $d_i:X_{n+1} \rightarrow X_n$ for each $i$ with $0 \leq i \leq n+1$ such that $d_id_j = d_{j-1}d_i$ if $i<j$.
A simplicial set consists of a sequence of sets $X_0,X_1 \cdots$ and for each $n \geq 0$, functions $d_i:X_{n} \rightarrow X_{n-1}, s_i:X_n \rightarrow X_{n+1}$ for each $i$ with $0 \leq i \leq n$ such that \begin{aligned} & d_id_j = d_{j-1}d_i ~\text{if}~ i<j \\ & s_is_j = s_{j+1}s_i ~\text{if}~ i \leq j \\ & d_is_j = s_{j-1}d_i ~\text{if}~ i<j \\ & d_js_j = d_{j+1}s_j=id \\ & d_is_j = s_jd_{i-1} ~\text{if}~ i>j+1 \\ \end{aligned}
In this regard, if $A$ is a simplicial set, take the same sequence $\lbrace X_n \rbrace$ and the same $\lbrace d_i \rbrace$, now $A$ become a delta set! This suggests that every simplicial set is a delta set, which is not true!
What goes wrong here?
This has already been answered fully in comments but we should take this question off the unanswered queue.
$\newcommand{\set}{\mathsf{Set}}\newcommand{\op}{{^{\mathsf{op}}}}$Nothing "goes wrong here". It's (almost exactly) like the difference between a function and its restriction; if I have some $f:\Bbb R\to\Bbb R$, it's still true that I have a function $[0,1]\to\Bbb R$ given by restricting $f$. This restriction just has less information. Any simplicial set is a delta set once you forget the $s_\bullet$ degeneracy maps.
Why do I say this analogy is so close? If you understand a bit about the language of categories, we can make the analogy precise (simplicial sets are often discussed in a categorical setting anyway, this analogy is not just a useless example!)
Let $\Delta$ denote the finite nonempty ordinal category: its objects are $[n]=\{1,2,\cdots,n\}$ where $n\in\Bbb N$ and its arrows are the weakly monotone functions, regarding each $[n]$ as ordered by $1\le 2\le\cdots\le n$. Let $\Delta_+$ denote the wide subcategory of $\Delta$ given by all the same objects but only the strictly monotone functions.
A simplicial set is exactly a functor $\Delta\op\to\set$. A semi-simplicial set is exactly a functor $\Delta_+\op\to\set$ (a.k.a "delta set"). Given any simplicial set $X:\Delta\op\to\set$ I can restrict it (like a function) along the natural inclusion $\Delta_+\op\to\Delta\op$ to find a delta set $X':\Delta\op\to\set$.
Like functions, we can also, in lucky situations, extend these functors. It turns out that any delta set $Y:\Delta_+\op\to\set$ has a 'free' extension to a simplicial set $X:\Delta\op\to\set$.
At the end of the day, by a very standard abuse of language, it is true that any simplicial set is a delta set.