There are certainly links between the two; a good place to start would be looking at the theory for $GL_2$ and $SL_2$. Henniart's appendix to Breuil-Mezard's Multiplicités modulaires et représentations de $GL_2(Z_p)$ et de $Gal(\bar{Q_p}/Q_p)$ en $l=p$ makes explicit the construction for $GL_2$ via strata, and it's simple enough to see that you can do more or less the same thing for $SL_2$ (now requiring your stratum to be simple $[\mathfrak{A},n,0,\beta]$ with $\beta\in\mathfrak{sl}_2$), while Nevins' Branching rules for supercuspidal representations of $SL_2$ gives an introduction to Yu's construction specified somewhat to the case of $SL_2$.
In this case, it's pretty clear that both constructions are doing more or less the same thing. Yu starts with a "tamely ramified cuspidal $G$-datum" $(T,y,r,\phi)$, i.e. a quadruple where:
$T$ is an anisotropic torus of $SL_2$, i.e. the group of points in $E^\times\cap SL_2$ for $E/F$ a quadratic extension embedded into $\mathrm{Mat}_2(F)$ in a suitable way;
$y$ is a point in the building of $SL_2$, where it turns out that we only ever need three points, $0$, $1/2$ and $1$ in order to determine the necessary parahorics;
$r$ is (essentially) the level of the supercuspidal that you're constructing;
$\phi$ is a "generic" character of $T$.
Yu then defines a suitable character $\psi$ on a group $H$ arising from the Moy-Prasad filtration, such that $\psi$ agree with $\phi$ on $T\cap H$, and extends this to a larger group from which the supercuspidal is induced.
On the other hand, here's the approach via strata. You start with a "simple stratum" $[\mathfrak{A},n,0,\beta]$, where:
$\mathfrak{A}$ is a hereditary $\mathfrak{o}_F$-order, which is essentially defining which parahoric we use, just as the $y$ in Yu's construction did;
$n$ is (essentially) the level of the supercuspidal;
$\beta$ defines a quadratic extension $F[\beta]/F$, as well as a character $\psi_\beta$ of a group $H$ in the filtration of the parahoric $\mathfrak{A}^\times$.
Then, as before, you extend, this time going from your filtration subgroup $H$ to the product $TH$, where $T=F[\beta]^\times$ is an anisotropic torus, then extend in to the same larger group from which Yu induced the supercuspidal (Heisenberg and $\beta$-extensions, although for $SL_2$ you can be more naive and just choose any suitable extension in exactly the same way as Henniart does for $GL_2$).
Both of these constructions look very similar: they specify more or less the same information about your supercuspidal in slightly different forms, and they end up inducing the supercuspidal from the same group $TH$ for $T$ an anisotropic torus and $H$ a filtration subgroup. The main difference is that Yu begins by defining the representation on $T$, while Bushnell-Kutzko begin by defining the representation on $H$ (or at least on some slightly smaller group contained in $H$).
