Integrality gap and approximation ratio
There are various examples in which a semidefinite programming relaxation allows an approximation that is superior to known integrality gaps for linear programming relaxations.

For example, the standard linear programming relaxation of max cut has an integrality gap of 1/2, and this is true even for much more sophisticated linear programming relaxations (c.f. de la Vega-Kenyon and Schoenebeck-Trevisan-Tulsiani), but the Goemans-Williamson SDP algorithm has approximation .878...

The Leighton-Rao linear programming relaxation of sparsest cut has an integrality gap $\Omega(\log n)$, but the Arora-Rao-Vazirani SDP algorithm has approximation $O(\sqrt{\log n})$.

Perhaps less well known, Karloff and Zwick shows that using SDP one can approximate Max 3SAT, in the version in which clauses can have 1, 2 or 3 literals, within 7/8, while Goemans and Williamson had studied a linear programming relaxation that they used to prove a 3/4 approximation (Yannakakis had given 3/4 approximation earlier by other methods), and the Goemans-Williamson LP relaxation of Max 3SAT is easily seen to have integrality gap 3/4.