I learn dg categories from Toën's lecture notes Lectures on dg-categories, where the tensor product $T\otimes T'$ of two dg-cateogries is defined to be such that it has objects of pairs $(x,x')$ and mapping spaces as $$(T\otimes T')\big((x,x'),(y,y')\big)=T(x,y)\otimes T'(x',y')$$ This definition makes the category $\mathrm{dgCat}$ of dg-categories form a closed symmetric monoidal category. And passing to the homotopy category $\mathrm{Ho}(\mathrm{dgCat})$, the derived tensor product $-\otimes^{\mathbb{L}}-=Q(-)\otimes-$ makes it also form a closed monoidal category.
But there is also a concept of tensor product of $\infty$-categories described in Lurie's long book Higher Algebra Section 4.8, which comes from the symmetric monoidal structure on $Pr^{L}$ the $\infty$-category of presentable $\infty$-categories. So for two dg-categories we have Lurie's tensor product donoted by $T\otimes^{L}T'$. It is characterized by the property that continuous functors (preserving colimits) from $T\otimes^L T'$ to $C$ are given by bilinear functors $T\times T'\rightarrow C$ i.e. functors which are continuous in each variable.
Should the two tensor products $\otimes^{\mathbb{L}}$ and $\otimes^L$ be the same? And can we describe Lurie's tensor product more concretely (writing down its objects and mapping spaces)?
No, these are not the same. There is an explicit description of the Lurie tensor product, given in HA, Prop. 4.8.1.17: given $\mathscr{C},\mathscr{D}\in\mathrm{Pr}^L$, their Lurie tensor product is given by $\mathscr{C}\otimes^L\mathscr{D}\simeq\mathrm{RFun}(\mathscr{C}^\mathrm{op},\mathscr{D})$, where $\mathrm{RFun}(-,-)$ denotes the full subcategory of the functor category on right adjoint functors. The functoriality of this object in $\mathrm{Pr}^L$ is admittedly a bit annoying to describe, and is easier in $\mathrm{Pr}^R$, but in general you have to switch between right adjoint and the corresponding left adjoint functors at appropriate times to define the functoriality of $\mathrm{RFun}(\mathscr{C}^\mathrm{op},\mathscr{D})$ in both terms.
Even without understanding functoriality, however, we can already see that $\mathscr{C}\otimes^L\mathscr{D}$ will in general have many more objects than just $\mathrm{Ob}(\mathscr{C})\times\mathrm{Ob}(\mathscr{D})$, (just like an ordinary tensor product of modules has many more elements than the product of the modules has). So the Lurie tensor product does not correspond to the derived tensor product in the case of dg-categories.
The latter derived tensor product actually just models the product of two $\infty$-categories (where of course the product is taken in $\mathrm{Cat}_\infty$). To see this, note that we can consider a dg-category $\mathcal{C}$ to be an $\infty$-category by hitting its chain complex enrichment with the Dold-Kan correspondence (which is lax monoidal) to turn $\mathcal{C}$ into a category $\mathcal{C}'$ enriched over simplicial abelian groups. Every simplicial abelian group is a Kan complex, so $\mathcal{C}'$ can also be considered as a category enriched over Kan complexes. Hitting $\mathcal{C}'$ with the homotopy coherent nerve, we obtain an $\infty$-category $N^\mathrm{coh}\mathcal{C}'$, which is equivalent to any other standard way to turn dg-categories into $\infty$-categories. (I am taking here quasicategories as the main model for $\infty$-categories. For the argument below, we could also just stop once we have built a Kan-enriched category.)
However, the Dold-Kan correspondence is fully monoidal in a homotopical sense (HA, Prop. 1.2.3.28), so given two dg-categories $\mathcal{C}$ and $\mathcal{D}$, the mapping Kan complexes of $(\mathcal{C}\otimes\mathcal{D})'$ are homotopy equivalent to the products of the mapping Kan complexes of $\mathcal{C}'$ and $\mathcal{D}'$. In other words, the Kan-enriched categories $(\mathcal{C}\otimes\mathcal{D})'$ and $\mathcal{C}'\times\mathcal{D}'$ (the latter product taken in the $1$-category of simplicially enriched categories) are Dwyer-Kan equivalent. As a consequence, the associated $\infty$-categories $N^\mathrm{coh}(\mathcal{C}\otimes\mathcal{D})'$ and $N^\mathrm{coh}\mathcal{C}'\times N^\mathrm{coh}\mathcal{D}'$ are also equivalent (the homotopy coherent nerve preserves products of $1$-categories because it is a right adjoint, and all products in sight are homotopy products as well because we are working with fibrant objects in the respective model categories, so we truly get an equivalence in $\mathrm{Cat}_\infty$).