Here attached is a good survey paper authored by Hamid Krim and Mats Viberg.

Generally speaking from one aspect, the difference is between the first order estimation (linearly estimation schemes, such as classic, Capon) and the second order estimation (high-order estimation schemes, such as MUSIC, root-MUSIC, ESPRIT). 

BTW, there traditionally is a "hard" limitation on the number of arriving signals M to be estimated by a sensor array of the size L. Something like M

You can open this link for any information about your problem http://www.vissta.ncsu.edu/publications/ahk/spm1996.pdf 

regards

Lubis

It depends on the array structure if you have a Uniform Linear Array (ULA) go for root-MUSIC, which is much less complex than MUSIC (requires an exhaustive research). If your array is arbitrary distributed go for MUSIC. 

As pointed out by colleagues, CAPON, and Bartlett beamformers perform less then MUSIC and root-MUSIC. You should also look for ESPRIT algorithm who provides an algebraic estimation.

According to the aforementioned valuable answers, suggested/attached files and links and given scenario I observed and conclude that for uniform linear array capon method is able to distinguish between two signals while the classical method failed to differentiate between them, but Capon’s method fails if other signals that are correlated with the signal of interest are present. On other hand MUSIC can resolve closely spaced signals which cannot be detected by Capon’s method while Root-MUSIC increase the resolution performance, especially at low SNR and decrease the computational complexity which is based on polynomial rooting..

There is also the Min-Norm method I consider better than MUSIC. See the paper in attach.