Обычно пишу обыкновенно.

Но какая разница между ними? Как употребляется и то и другое?

Обычно is more common. Обыкновенно is a bit higher style. Or archaic.
Необычный - uncommon, unusual, strange (neutral). Необыкновенный - special, phenomenal (with a strong emotional impact).

ОК, thanks. From now on I will usually use обычно, since I don't usually speak or write in high style!

Isn't the difference between those two similar to that between "usually" and "normally" in English?

Sometimes there can be a subtle difference. It's slightly more pronounced in adjectives "обычный-обыкновенный":
обычный - usual, common -> everyday, widespread, familiar
обыкновенный - usual, common -> ordinary, typical for this kind of things
обыкновенно - done in typical/habitual/expected/etc. manner

So "обыкновенно" does not necessarily mean high style. The most common example of colloquial usage is:
- Как ты это сделал? (about something obvious)
- Как-как.. Обыкновенно. (~duh! how do you think???)

Quote:

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Как стирать носки в стиралльной машинке?
Обыкновенно. Кидаешь носки в машинку. Засыпаешь порошок и вперед
Source: Ответы@Mail.Ru: как стирать носки в стиралльной машинке?

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It sounds high srtyle and archaic mostly when used instead of "usually", when you describe some regular activity or habits:
Я обыкновенно обычно пью чай по утрам.
Он обыкновенно обычно не здоровается с незнакомыми людьми.

Quote:

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Originally Posted by gRomoZeka

So "обыкновенно" does not necessarily mean high style. The most common example of colloquial usage is:
- Как ты это сделал? (about something obvious)
- Как-как.. Обыкновенно. (~duh! how do you think???)

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I wanted to write something like that but I feel like such phrases are rather rare and tend to go out of common language with time.

Quote:

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...and tend to go out of common language with time.

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This time has not come yet. :) "Обычно" is more universal and safe to use nowadays, but in situations like this it can't replace "обыкновенно". One can try, of course, but it will sound a little weird.

Quote:

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Originally Posted by gRomoZeka

Sometimes there can be a subtle difference. It's slightly more pronounced in adjectives "обычный-обыкновенный":
обычный - usual, common -> everyday, widespread, familiar
обыкновенный - usual, common -> ordinary, typical for this kind of things
о

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Please could you give a couple of examples to illustrate this. Are you saying that there are occasions when replacing обычно by obyknovenno is impossible, even in more formal styles?