Dough, in some cases, the Higgs and confinement phases are not separated at all and form the total screening phase, see Banks Rabinovici 1979:
and Fradkin Shenker 1979:
More recently, those and similar relationships were studied in the context of supersymmetric field theories and supersymmetric string vacua. The papers by Seiberg and Witten for $N=2$ were the main progress. See also papers such as this random one:
In some cases, the confinement and Higgs phases may be distinct. The relationships between the two phenomena may be understood as a sort of S-duality - a strong-weak duality or a generalized electro-magnetic duality. For example, the last paper I quoted explains why confinement may be understood as condensation of monopoles (the dual, "magnetically" charged particles, which is kind of the S-dual of the normal Higgs mechanism which gives vevs to "electrically" charged fields).
Supersymmetric gauge theories are full of S-dualities, much like stringy vacua.