Looking for an apartment in bogota

No difference for a foreign renter as for a native renter.

BUT both may be required to have one or more fiadores (guarantors) to assure the landlord that the rent will be paid by someone, and some agencies may also require a cédula de extranjería or national ID card for a foreigner, just as a cédula is required for a native.  Many realize that a foreign passport serves the same purpose, but some will not accept it.

If you are dealing directly with the owner none of that may be required by them - but some may still want you to pay several months' rent up front, even a whole six months' rent.  Some may not.

This is why it may pay to just walk around and look for the Apartamento en Arriendo and Se Arrienda signs in the windows.  As cccmedia has suggested it may also pay to notify the celadores and porteros of various apartment buildings that you're looking to rent and would pay them a finder's fee if you get a rental, say as much as 50,000 to 100,000 COP (about 17 to 34 USD) which is a pretty good chunk of change for people who may earn only the minimum monthly wage of 737,717 COP (a bit over 250 USD at today's exchange rate).