Finite Element: volumetric integrals, internal polynomial order
Classical finite element methods assume continuous or weakly continuous approximation spaces and ask for volumetric integrals of the weak form to be satisfied. The order of accuracy is increased by raising the approximation order within elements. The methods are not exactly conservative, thus often struggle with stability for discontinuous processes.
Finite Volume: surface integrals, fluxes from discontinuous data, reconstruction order
Finite volume methods use piecewise constant approximation spaces and ask for integrals against piecewise constant test functions to be satisfied. This yields exact conservation statements. The volume integral is converted to a surface integral and the entire physics is specified in terms of fluxes in those surface integrals. For first-order hyperbolic problems, this is a Riemann solve. Second order/elliptic fluxes are more subtle. Order of accuracy is increased by using neighbors to (conservatively) reconstruct higher order representations of the state inside elements (slope reconstruction/limiting) or by reconstructing fluxes (flux limiting). The reconstruction process is usually nonlinear to control oscillations around discontinuous features of the solution, see total variation diminishing (TVD) and essentially non-oscillatory (ENO/WENO) methods. A nonlinear discretization is necessary to simultaneously obtain both higher than first order accuracy in smooth regions and bounded total variation across discontinuities, see Godunov's theorem.
Both FE and FV are easy to define up to second order accuracy on unstructured grids. FE is easier to go beyond second order on unstructured grids. FV handles non-conforming meshes more easily and robustly.
Combining FE and FV
The methods can be married in multiple ways. Discontinuous Galerkin methods are finite element methods that use discontinuous basis functions, thus acquiring Riemann solvers and more robustness for discontinuous processes (especially hyperbolic). DG methods can be used with nonlinear limiters (usually with some reduction in accuracy), but satisfy a cell-wise entropy inequality without limiting and can thus be used without limiting for some problems where other schemes require limiters. (This is especially useful for adjoint-based optimization since it makes the discrete adjoint more representative of the continuous adjoint equations.) Mixed FE methods for elliptic problems use discontinuous basis functions and after some choices of quadrature, can be reinterpreted as standard finite volume methods, see this answer for more. Reconstruction DG methods (aka. $P_N P_M$ or "Recovery DG") use both FV-like conservative reconstruction and internal order enrichment, and are thus a superset of FV and DG methods.