Learn Polish: Jeżeli zdasz ten egzamin, będziesz mieć szansę. - If you pass this exam, you will have a chance.

No, they mean the exact same thing ('if') and are completely interchangeable in this sentence. Jeśli is a bit shorter and more common in everyday speech, while jeżeli can sound slightly more formal, but you can safely use whichever you prefer.

English uses the present tense ('If you pass...'), but what tense is zdasz?

In Polish, if the condition refers to the future, you must use the future tense in the 'if' clause. Zdasz is the future tense of zdać (to pass). Where English says 'If you pass', Polish literally says 'If you will pass'.

Why is the first future tense just one word (zdasz) while the second is two words (będziesz mieć)?

This is because of verb aspect, which tells us if an action is completed or ongoing. Zdać (to pass) is a perfective verb for a one-time, completed action, so its future tense is a single word (zdasz). Mieć (to have) is imperfective for an ongoing state, so it uses the compound future tense: the helper verb będziesz plus the infinitive mieć.

Both egzamin and szansę are direct objects. Why does szansa change to szansę but egzamin stays the same?