In enriched category theory, weighted limits may be strictly more general than conical limits, in the sense that an enriched category with all conical limits may fail to have all weighted limits. However, in ordinary category theory (and $(\infty, 1)$-category theory too) weighted limits are the same as limits: any weighted limit can be rewritten as a conical limit in a uniform way.

Here is an analogy which might help you understand the meaning of weighted limits: if you have two elements of a group, say $x$ and $y$, you could certainly multiply them together to get another element $x y$... but you could equally well multiply them in the reverse order and get $y x$, or indeed any number of other combinations, such as $y^{-1} x^{-1}$ or $x y x^{-1} y^{-1}$ or whatever; the free group on two elements parametrises all the possible natural operations on two elements of a group. In the same way, a weight $\mathcal{I} \to \textbf{Set}$ parametrises all the possible natural operations on a diagram $\mathcal{I} \to \mathcal{C}$ where $\mathcal{C}$ is a complete category, because the category $[\mathcal{I}, \textbf{Set}]^\textrm{op}$ is the free complete category generated by a diagram of shape $\mathcal{I}$.

Specifically, if $\mathcal{C}$ is complete and $F : \mathcal{I} \to \mathcal{C}$ is a diagram, there is a unique (in the appropriate sense) limit-preserving functor $\{ {-}, F \} : [\mathcal{I}, \textbf{Set}]^\textrm{op} \to \mathcal{C}$ such that the composite with the Yoneda embedding $\mathcal{I} \to [\mathcal{I}, \textbf{Set}]^\textrm{op}$ is the given $F : \mathcal{I} \to \mathcal{C}$. Characterising $\{ W, F \}$ explicitly for each individual weight $W : \mathcal{I} \to \textbf{Set}$ leads to the "official" definition of weighted limits, which can then be applied to categories $\mathcal{C}$ not assumed to be complete. Writing each object in $[\mathcal{I}, \textbf{Set}]^\textrm{op}$ as a limit of a diagram built from the representables translates into a uniform way of writing weighted limits of any diagram $\mathcal{I} \to \mathcal{C}$ as conical limits of (other) diagrams in $\mathcal{C}$.

If you really insist on an analogy with real analysis, then perhaps I should warn that thinking of weighted limits as generalised limits is not helpful. It may be better to think of weighted limits as being analogous to functionals defined by integration against a measure ... but even this is a bit of a stretch, in my opinion.

In enriched category theory, weighted limits may be strictly more general than conical limits, in the sense that an enriched category with all conical limits may fail to have all weighted limits. However, in ordinary category theory (and $(\infty, 1)$-category theory too) weighted limits are the same as limits: any weighted limit can be rewritten as a conical limit in a uniform way.

Here is an analogy which might help you understand the meaning of weighted limits: if you have two elements of a group, say $x$ and $y$, you could certainly multiply them together to get another element $x y$... but you could equally well multiply them in the reverse order and get $y x$, or indeed any number of other combinations, such as $y^{-1} x^{-1}$ or $x y x^{-1} y^{-1}$ or whatever; the free group on two elements parametrises all the possible natural operations on two elements of a group. In the same way, a weight $\mathcal{I} \to \textbf{Set}$ parametrises natural operations on a diagram $\mathcal{I} \to \mathcal{C}$ where $\mathcal{C}$ is a complete category, because the category $[\mathcal{I}, \textbf{Set}]^\textrm{op}$ is the free complete category generated by a diagram of shape $\mathcal{I}$.

Specifically, if $\mathcal{C}$ is complete and $F : \mathcal{I} \to \mathcal{C}$ is a diagram, there is a unique (in the appropriate sense) limit-preserving functor $\{ {-}, F \} : [\mathcal{I}, \textbf{Set}]^\textrm{op} \to \mathcal{C}$ such that the composite with the Yoneda embedding $\mathcal{I} \to [\mathcal{I}, \textbf{Set}]^\textrm{op}$ is the given $F : \mathcal{I} \to \mathcal{C}$. Characterising $\{ W, F \}$ explicitly for each individual weight $W : \mathcal{I} \to \textbf{Set}$ leads to the "official" definition of weighted limits, which can then be applied to categories $\mathcal{C}$ not assumed to be complete. Writing each object in $[\mathcal{I}, \textbf{Set}]^\textrm{op}$ as a limit of a diagram built from the representables translates into a uniform way of writing weighted limits of any diagram $\mathcal{I} \to \mathcal{C}$ as conical limits of (other) diagrams in $\mathcal{C}$.

If you really insist on an analogy with real analysis, then perhaps I should warn that thinking of weighted limits as generalised limits is not helpful. It may be better to think of weighted limits as being analogous to functionals defined by integration against a measure ... but even this is a bit of a stretch, in my opinion.