"When (it is) compared with other mighty rivers of the world, the River Thames may appear small." There are two clauses in this sentence. The second one is the main clause and has the main subject -"the River Thames" - and the main (finite) verb - "may appear". The first clause is a subordinate clause. It is a dependent clause because it makes sense only with the second clause. More specifically, it gives more information about "the River Thames". Dependent clauses can include reduced verb forms like "compared", in this case. The words "it is" are implied and can be included, but they do not need to be included. In fact, it is good style to omit them.

It might help if you look up "participle clauses". That's the grammar for the first part of the sentence.

Strictly there are differences between 'compared to...' and 'compared with...' To compare to is to point out or imply resemblances between objects regarded as essentially of a different order. To compare with is mainly to point out differences between objects regarded as essentially of the same order. In actual use, I rarely see/hear these forms used correctly anyway. So, taking your sentence: ''When compared with other mighty rivers of the world, *the* River Thames may appear small.' But you could also use other forms, as you imply: 'When compared with other mighty rivers, the Thames may seem small,' for example. BTW: to compare is a verb and compared is its participle. This is quite a useful pattern in English.