"Громко" is used just to describe how loud you do something, speaking included. "Вслух" is used to describe the process of making something known to others by speaking.

I would interpret "I like to speak Russian aloud" as "Я люблю/Мне нравится говорить по-русски громко/вслух" depending on the context of situation. Your second sentence will be interpreted the same by nine out of ten people. "Громко" is to specific to be said in this case.

Examples of usage:

Скажи вслух (as opposed to "Скажи про себя").
Скажи громко (as opposed to "Скажи тихо").

Я бы не хотел, чтобы его мысли прозвучали вслух ("This is inconvenient for me to see him go public").
Эти слова он произносил громко ("The words were spoken aloud").
Чтение/читать вслух полезно для дикции ("Reading aloud is useful for your articulation").
Здесь читать громко нельзя, соблюдайте тишину. ("Reading aloud is not allowed here, so keep quiet").