The main difference is vocabulary and the verb system. Some classical words have different meanings in modern colloquial (e.g. classical semchen
"sentient being", just means "animal" in modern Tibetan), and there are many new words. The classical Tibetan verb system only has 3 tenses and an imperative, modern Tibetan uses a large number of auxiliary and modal verbs to distinguish many different aspects
Modern written Tibetan isn't the same as classical Tibetan. Modern namthars & commentaries are written in classical dharma language, but newspapers, formal announcements and letters, etc. are written in modern literary Tibetan, which is roughly halfway between spoken & classical Tibetan.
Modern colloquial gives you access to oral teachings, classical gives you access to texts. If you want to study at a shedra in India or Nepal, you will need to become
fluent in both, but it's not necessary to start
Of course, more and more lamas speak English, and more and more texts are translated. Unless you plan on living in a Tibetan community in Asia, I'd say that learning colloquial is less useful. I rarely have problems with lamas teaching directly in English or oral interpreters, whereas---despite the best efforts of excellent translators---English translations of Tibetan texts are often very confusing.
If you often find yourself staring at a sentence in an English translation and wondering what the hell it's trying to say, learn classical. If you often find yourself at a teaching wondering if you got the point the lama was trying to get across, learn colloquial.
Either of the two is a great help in learning the other. The script, the syntax, the particles, and a great deal of the vocabulary are the same in both. It's roughly comparable to the difference between the Canterbury Tales and modern spoken English, if English were still spelled exactly the same.