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The distinction lies in the quantifier order. For pointwise convergence, one first fixes a specific point $x$, then chooses $\epsilon>0$, and finds an index $N$ that may depend on both $\epsilon$ and $x$.

Conditions: A sequence of functions $f_n$ and a limit function $f$ are defined on a region.; Pointwise convergence: $\forall x, \forall \epsilon>0, \exists N$ such that $n\ge N \implies |f(x)-f_n(x)|<\epsilon$.; Uniform convergence: $\forall \epsilon>0, \exists N$ such that $n\ge N \implies |f(x)-f_n(x)|<\epsilon$ for all $x$ in the region.

The distinction lies in the quantifier order. For pointwise convergence, one first fixes a specific point $x$, then chooses $\epsilon>0$, and finds an index $N$ that may depend on both $\epsilon$ and $x$.

Conditions: A sequence of functions $f_n$ and a limit function $f$ are defined on a region.; Pointwise convergence: $\forall x, \forall \epsilon>0, \exists N$ such that $n\ge N \implies |f(x)-f_n(x)|<\epsilon$.; Uniform convergence: $\forall \epsilon>0, \exists N$ such that $n\ge N \implies |f(x)-f_n(x)|<\epsilon$ for all $x$ in the region.