Yes, there is. Even though the two verbs can be used interchangeably in many contexts, there is a difference in meaning, which I will explain in this post.
Besuchen is best translated as to visit or attend because it can be used to visit people and places as well as attend school, a course or university lecture or seminar. By contrast, besichtigen is translated as to visit in the specific sense of sightseeing. In short, it can only be used for places.
Ich will schon seit langer Zeit Rio de Janeiro besuchen,
(I have been wanting to visit Rio de Janeiro for a long time)
Normalerweise besucht sie ihre Freunde sehr oft.
(Normally, she visits her friends often)
Er besucht einen Deutschkurs.
(He attends a German course)
Du besuchst eine Universitätsvorlesung.
(You attend a university lecture)
Let’s now look at some examples for “besichtigen”:
Ich möchte das Castelo de São Jorge in Lissabon besichtigen.
(I’d like to visit Castelo de São Jorge in Lisbon)
Wenn man nach Köln reist, sollte man den Dom besichtigen.
(When you travel to Cologne, you should visit the Dom)
Viele Touristen in London besichtigen The Tower of London
(Many tourists in London visit the Tower of London)
As you can see from examples, the two German words can sometimes be taken to mean the same, but when I say “Ich möchte Rio besuchen” the emphasis is on visiting the place more generally, while “besichtigen“ here would only make sense with a tourist sight I would like to see.
Ich möchte Cristo de Rentor in Rio besichtigen.
By the way, the German word for sight is “die Sehenswürdigkeit”, which literally means thing worth seeing. Nice word, isn’t it? There are many other long German nouns that are really interesting compound nouns.
On our German Language Blog “Auf Deutsch, bitte!”, you will find posts that clarify many other German words, such as kennen vs. wissen, am Morgen vs. Morgens, false friends in German and English like "blenden" and "blend" , become vs. bekommen and other false cognates, and how to say something is fun in German. We also have posts on many topics in German grammar, such as German word order in general, and conjunctions such as dass in German, we explain the passive voice in German and tenses like the Präteritum and the Plusquamperfekt, as well as the most important grammar terms and what they mean.