An adjacency matrix is a square matrix utilized to represent a finite graph, whereas an adjacency list consists of a collection of unordered lists serving the same purpose.

An adjacency matrix is a two-dimensional array of size $V \times V$, where $V$ denotes the number of vertices in the graph. The entries of this matrix indicate whether pairs of vertices are adjacent. Each vertex has a corresponding list that outlines the vertices it is directly connected to. The size of the matrix is $V^2$, making it particularly suitable for dense graphs—those in which the number of edges is close to $V^2$.

In contrast, an adjacency list comprises separate lists, with each list detailing the set of neighbors for a specific vertex. This structure is more space-efficient for sparsely connected graphs, where the number of edges is significantly less than $V^2$. In an adjacency list, each vertex maintains its own list of connected vertices, resulting in a total size of $2E$, where $E$ represents the number of edges.

When considering time complexity, the performance of various operations differs between the two representations. For instance, checking whether an edge $(u, v)$ exists in the graph requires $O(1)$ time for an adjacency matrix, while it takes $O(V)$ time for an adjacency list. However, when it comes to iterating over the edges, the adjacency matrix incurs a cost of $O(V^2)$, whereas the adjacency list operates in $O(V + E)$ time.

In conclusion, the decision to use an adjacency matrix or an adjacency list hinges on the characteristics of the graph and the specific operations to be performed. If the graph is dense and edge lookups are frequently needed, an adjacency matrix is preferable. Conversely, if the graph is sparse and edge iteration is a common operation, an adjacency list is the more efficient choice.