The definition of subnet is more convoluted than expected. The idea seems to be that the definition is such that the equivalence compactness $\Leftrightarrow$ every net has a convergent subnet holds.
In here Henno Brandsma states that for the equivalence to hold we cannot define a subnet of a net $(x_i)_{i\in I}$ to be a net $(x_i)_{i\in J}$ where $J$ is cofinal in $I$. Let $(P)$ be the property that every net admits a convergent subnet in the (non-standard) sense above.
Onwards let subnet have its classical definition. Clearly in any topological space $(P)$ implies that every net has a converging subnet. So the comment implies that there are spaces that are compact but still not satisfy $(P)$.
What would one such space look like?
$X = \{0,1\}^I$ in the product topology, where $I = 2^{\mathbb{N}}$ (or any other index set of size $\mathbb{c}$) is an example of a compact Hausdorff space, which has a sequence (i.e. a net with index set $\mathbb{N}$ in the usual order) $(x_n)$ without a convergent subsequence (I showed that here). $\beta \omega$ is also a classical example , but takes a bit more theory to define, also using sequences. Note that a subsequence is just the subnet determined by a cofinal subset of $\mathbb{N}$. So such an $(x_n)$ is a net witnesssing that $X$ does not obey $(P)$. It's because of the insufficiency of subsequences of sequences (and taking larger index sets and larger products we can also, I think, find examples for some uncountable index sets and their cofinal subsets, or use this example etc.) that the more complex definition of subnet as used in Kelly and Engelking etc., came to be defined and studied.