The main difference to these two phrases is that "I believe" gives your sentence more emphasis, i.e. you feel more sure about your statement. For "I suppose" you are placing less feeling in your sentence. So, while you may have the general idea that the sentence is happening, you are not very sure of yourself. Long story made short: "I believe" is a slightly stronger sounding statement than "I suppose". Hope this helps!

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'I believe' and 'I suppose' can mean almost the same thing when you want to express an opinion. We use them both to say something is probable or likely although 'suppose' is more uncertain than 'believe'. We also use suppose to presume or assume something to be true without certain knowledge. 'I suppose he meant to kill her'.