To what extent was Renaissance "natural philosophy" similar to modern science? 

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Like modern science, the "natural philosophy" of the Renaissance was derived largely from observation. It is well-known that Renaissance humanist thought was deeply informed by classic texts, and many of the most famous intellectuals of the Renaissance, including Leonardo and Galileo (to focus only on the Italian Renaissance) criticized older, accepted systems, particularly the Ptolemaic model of the universe, that did not comport to observations. Indeed, the fundamental shift in thinking that is often associated with a "scientific revolution" was part of the Renaissance, namely a new, more empirical approach to understanding the world as opposed to using reason alone. Francis Bacon and Galileo were especially important in applying inductive reason informed by observation and even experimentation to scientific inquiry, and it was no coincidence that major scientific advances tended to accompany technological developments, like the telescope and the micoscope. 

There were many differences between modern science and the way that Renaissance intellectuals understood natural philosophy. One is that mainstream science today is regarded as a thoroughly secular institution and worldview. Early modern thinkers did not view the world this way, rather viewing science as a way to grasp the mysteries of the world. Science was also less institutional, and, from a modern perspective, non-academic and non-specialized. There were no jobs, in other words, in science during the Renaissance, and apart from royal or noble patronage, no government involvement or support.