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As I understand prohibition of both is under the basis that it could lead to them being repaired. However there are poskim that consider riding a bike permissible.

Why would it be okay to ride a bike but not play an instrument with this in mind?

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There is a fundamental difference between the concern of repairing an instrument and the concern of repairing a bicycle. The former is mentioned in the Talmud:

Beitza 36b

ולא מטפחין ולא מספקין ולא מרקדין: גזרה שמא יתקן כלי שיר

NOR CLAP THE HANDS, NOR SLAP THE THIGHS, NOR DANCE; it is a preventive measure lest he might repair musical instruments.

(Soncino translation)

A principle often mentioned in halachic writings about new situations is that we do not invent our own preventative decrees. If the Sages were concerned about something they issued a decree about it. If they didn't issue a decree about it then we don't have to invent our own. Indeed, R. Joseph Hayyim of Baghdad, in his responsum on bicycles explicitly makes this point:

Shu"t Rav Pe'alim O.C. 1:25

I have heard it further said that we should be concerned that there will be in this some break on Shabbat and he will come to fix it. And this too is nonsense, and does not deserve to be said. First, because this object is not prone to breaking, and furthermore we do not invent decrees on our own that were not decreed by the Sages. For according to this there are many things that are permitted to use but are prone to breaking that we would need to forbid because of this decree. And we find many things that the Sages forbade on Shabbat lest one come to write or to fix, and that notwithstanding we don't generate these decrees on our own by other things that they did not delineate these decrees. And therefore there are many things where the decree of "lest he come to write" or "lest he come to fix" would be applicable but they are nevertheless permitted, because the Sages did not make this decree for them.

And see what the Birchei Yosef writes in the name of Maharam Provencial that it is permitted to compose a song by mouth with rhythm [?] and we do not decree "lest he come to write", because we only have that which is explicit in the words of the Sages, end quote.

And so wrote Tosafot in Chulin, and these are their words: "We do not compare the decrees of the Sages to each other except where the Talmud does so", end quote.

And see the Shach in Yoreh Deiah where he wrote that we do not compare decrees to each other as Tosafot wrote in Chulin, see there.

And so wrote the Magen Avraham, see there.

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