Bowing on a violin is important: Playing downbow will start closer to the end of the bow where your hand is (the frog), which will allow your to transfer force onto the strings quite directly, allowing for strong and expressive playing. Playing upbow will start closer to the end that is away from the hand (the tip), which will give a lighter sound.
Now, a modern bow is weighted so that it is balanced quite far towards the middle, while on baroque bows you tend to have more mass towards the frog and less towards the tip, which will increase the effect of downbow and upbow massively. But even on a modern bow playing downbow will allow you put stronger accents and emphases on the notes.
This means that passages can get awkward to phrase if you reverse the bowing because you need to put in more bow control to get an accent on upbow or to avoid accenting a downbow.
Furthermore playing downbow is a little bit "easier", as it requires you lowering your lower arm while upbow requires you to lift it. This means that playing downbow will require less effort to get a big, fluent motion. On the other hand the upward motion makes a controlled spiccato (that is the bow shortly leaving the string after each note) easier.
Starting on particular bowing will affect the bowing of the following notes, unless you change bowing at some point, so it strongly how a passage is phrased and articulated, especially with baroque bows.