Biblical and Modern Israeli Hebrew: What's the Difference?

The short answer is that Modern Israeli Hebrew is surprisingly similar to Biblical Hebrew, and in many ways it's even easier to learn. For an Israeli reading the Torah is like reading Hamlet for an English speaker. However if you could magically transport an Israeli into biblical times or bring back Moses into modern day Israel, both would have trouble just sitting and shmoozing with other people.

What's similar?

The Alephbet — Although ancient Hebrew used different shapes for the letters, the style of calligraphy used in the Torah is compatible with Israeli Hebrew.
They don't write vowels... except when they do! Ancient Hebrew didn't normally use vowels. The vowels and other markings were actually developed in ancient times to ensure people would pronounce prayers and bible passages correctly, and those same vowels are still used today to teach children or clarify pronunciation in dictionaries.
Most basic words! Simple phrases such as "this is my house" or "I love my mom" are said pretty much the same today as back then.

Israeli Hebrew is still very similar to ancient Hebrew because the Jewish exile and diaspora put the ancient language on ice. Hebrew was used for prayer, study and philosophical texts but not for everyday chitchat. It was Eliezer Ben-Yehuda who revived the language with a vision to have the Jews returning to Israel in the 1800's from Europe and Arab countries, all speaking different mother tongues, speak one common language, the language of the Jewish people. His greatest contribution to the making Hebrew a living language again was probably a dictionary that introduced new words for modern purposes, but based on the Hebrew of the bible and the large body of rabbinical works.

How is Modern Hebrew different from Biblical Hebrew?

There were many dialects of Ancient Hebrew. The biblical era spanned over a thousand years and many places. Even the Bible itself sometimes draws attention to how certain individuals or groups had different speech patterns. We have some idea of how people spoke, because Hebrew has been maintained by Jews as a liturgical and philosophical language for thousands of years, and because other related languages such as Arabic and Aramaic can give us clues, but we can't know for certain how anyone spoke in ancient times. The one thing we know is that there were many sub-dialects and most likely a lot of variations in exactly how each consonant or vowel would have been pronounced.

Fewer sounds (vowels and consonants) — Ancient Hebrew differentiated more carefully between vowel sounds. Modern Hebrew treats the different "ah" and "eh" sounds as basically the same. Originally Aleph and Ayin had very distinct sounds, and sometimes Ayin may have had something like a G sound. Tav in Israeli Hebrew has just one sound, but in Ashkenazic Hebrew it can sound like T or S, and at least some Israelites probably pronounced it as "TH." And Resh was probably pronounced more like in Spanish (as Sefardi Jews say it) whereas Israeli Hebrew uses a sound closer to the French R, which is probably similar to the original Ayin sound. But these pronunciations also varied by region and over time.
Grammar — A fairly big difference is that ancient Hebrew used verb tenses differently, with a richer range of options. One example is the Vav Hahipuch.
Vav Hahipuch — The Vav Hahipuch is a neat feature of biblical Hebrew, where the presence of a Vav at the beginning of a verb conjugation actually changed the tense of the verb. Normally that Vav actually just means and.
Israeli words for modern technology — Israeli Hebrew has sometimes used old words for new things, but has both invented and borrowed words from other languages to describe new technology and modern scholarship. The word for car (מְכוֹנִית) is adapted from a rare Biblical word, whereas the word for technology (טֶכנוֹלוֹגִיָה) comes from Russian.
Writing: cursive and typefaces — Although the writing was listed above as one of the things that's similar, Israeli Hebrew has more writing styles, including modern typefaces (eg. fonts) and a cursive handwritten style.