It seems to me that they are different (I had the same question, and was searching online to see if I messed up somewhere).

To see that they are not the same note that for any given $U_{N,\mathfrak{a}}$ and for any $m\gg 0$, the element $p^{-m}[x]$ is a nonzero element of $U_{N,\mathfrak{a}}$ for any nonzero $x\in \mathfrak{a}^{p^m}$. On the other hand, $p^{-n} W(\mathfrak{a})+p^NW(R)\subset p^{-n}W(R)$ does not have an element of this form when $m>n$. Thus this latter set cannot contain a set of the form $U_{N,\mathfrak{a}}$.

In fact, I cannot even figure out why $W(R)[\frac 1p]$ is a topological ring in the Brinon-Conrad topology... In the linked exercise, it is claimed that $U_{N,\mathfrak{a}}\cdot U_{N,\mathfrak{a}}\subset U_{N,\mathfrak{a}}$. But I don't think this is true. Let $\varpi\in A$ be a pseudouniformizer. Letting $N=0$ and $\mathfrak{a}=(\varpi)$, I believe that
\begin{equation}
p^{-n}W(\mathfrak{a}^{p^n})+p^NW(R)
\end{equation}
is the set of all elements that can be written as
\begin{equation}
p^{-n}[x_{n}]+p^{-n+1}[x_{n-1}]+\dots + p^{-1}[x_{1}]+a
\end{equation}
where $a\in W(R)$, and $x_{i}\in (\varpi^{p^i})$ for $i=n,\dots,1$.

If this is the case, then for any $\ell>0$, the element $p^{-\ell}[\varpi^{p^\ell}]$ is in $U_{N,\mathfrak{a}}$. But $p^{-\ell}[\varpi^{p^\ell}]\cdot p^{-\ell}[\varpi^{p^\ell}] = p^{-2\ell}[\varpi^{2p^\ell}]$. This is not of the form $p^{-2\ell}[x_{2\ell}]$ for some $x_{2\ell}\in (\varpi^{p^{2\ell}})$ since $2p^\ell<p^{2\ell}$ for all large $\ell$. So I do not think the claim is true.

(In a similar manner, I don't think the product of any two opens $U_{N,\mathfrak{a}}\cdot U_{M,\mathfrak{b}}$ is contained in $U_{0,(\varpi)}$: for any fixed $\alpha,\beta>0$, we have $(\alpha +\beta) p^\ell<p^{2\ell}$ when $\ell\gg 0$.)

Edit: Now I am also a bit worried that $p^{-n}W(\mathfrak{a})+p^NW(R)$ is not a basis for the inductive limit topology. If this were the case, then since multiplication by $p$ should be a homeomorphism on $W(R)[\frac 1p]$, we can multiply this set by say $p^{2n}$ to get an open subset $p^nW(\mathfrak a)+p^{N+2n}W(R)$. But if this were open in $W(R)[\frac 1p]$, then the inclusion $W(R)\subset W(R)[\frac 1p]$ would not be continuous. Open sets of $W(R)$ contain some $W(\mathfrak a)+p^NW(R)$ which always contain some nonzero Teichmuller element $[x]$ where $x\in \mathfrak a$ is nonzero, while the above set does not contain nonzero Teichmuller elements for $n>0$.

I think that the inductive limit topology should have a neighborhood basis of $0$ given by sets of the form
\begin{equation}
\bigcup_{n>-N} p^{-n}W(\mathfrak a_n)+p^NW(R)
\end{equation}
where the $\mathfrak a_n\subset R$ are open ideals. This differs from the Brinon-Conrad topology because the ideals $\mathfrak{a}_n$ can go to $0$ as slowly as necessary to get around the issue raised above.