If you check into dictionary,both definitions are the same.In reality,each term carries a different meaning in the bussiness world.
Amalgamate - combination or fusion of more than 1 entity.All participants' identities are lost.A new company is formed with a brand new corperate name.This new company is the fusion of all companies.
Merge - combination or fusion of more than 1 entity.One company is a stronger or host company and will maintain as the post-merging company identification.This means,all participants lost their identities except one company.This one company will be the new company.There are different types of merging terms.But that is for some other time.