Learn Russian: Мы часто вспоминаем об истории и о науке. - We often reminisce about history and about science.

Both words mean 'about' and require the Prepositional case. Just like 'an' and 'a' in English, you use об before a word starting with a vowel sound (like the и in истории) and о before a word starting with a consonant sound (like the н in науке).

Both история (history) and наука (science) are feminine nouns. Why do they end differently in this sentence?

They are both in the Prepositional case, but follow slightly different rules. Regular feminine nouns ending in -а change to -е (наука becomes науке). However, feminine nouns ending in -ия are a special group; their Prepositional ending is -ии (история becomes истории).

Does вспоминаем just mean 'remember'?

The verb вспоминать specifically means 'to recall', 'to bring back to mind', or 'to reminisce' (hence the prefix вс-, meaning 'up' or 'again'). If you simply want to say that a fact is stored in your memory, you would use the standard verb for 'remember': помнить.

Why is the word for 'about' repeated before both nouns? Couldn't we just say 'об истории и науке'?

In English we often drop the second preposition, but in Russian, when you have a list of nouns connected by и (and), it is standard and sounds much more natural to repeat the preposition before each item.