This is right up my alley since I'm a marine chemist, and I study iron limitation. In water, iron is present mainly as Fe(III), or ferric iron. This is because most aquatic environments are pretty oxidizing. Ferric iron is highly insoluble and so most of it is found in solid form, usually as precipitates that coat particles or other surfaces.

In acidic aquatic environments (in low oxygen waters, for example, or in mine shafts), iron is present mainly as Fe(II)or ferrous iron, which is much more soluble. In fact, ferric iron (Fe-III) is so insoluble in seawater and the concentration of dissolved iron is so low that it actually limits the growth of marine micro algae, a major source of oxygen and an important sink for carbon dioxide.

So, to get back to your question: Iron precipitates(ferric iron) are brown, while dissolved iron (ferrous iron) is clear. What is happening in your experiment is that when you add the iron nitrate to (neutral) water, the iron dissolves and is immediately oxidized and re-precipitates as a fluffy brown solid. When you add the concentrated nitric acid, the ferric iron precipitates dissolve and stay in solution as clear, ferrous iron.

Try adding some base, such as a few pellets of sodium hydroxide or a pinch of sodium carbonate (careful: the solution may give off heat, fizz and noxious fumes). Can you guess what you'll see?

Now, about the purple color: I guess I need more information. Iron, nitric acid, water and salicylates (e.g. aspirin) will form a purple color, with the amount of purple proportional to the amount of iron in your solution. Manganese precipitates are also purple.

Actually, what you saw when you first added the iron (III) solution to water was a slight hydrolysis of the iron (III) forming hydroxide and oxide-hydroxide precipitates. You can confirm this by just setting it aside for a few hours --- there will be a small amount of precipitated matter at the bottom.

The various equilibria are not easy to write, but this is as good as anything else:
Fe³⁺ + 2H₂O ---> Fe O(OH) (brown precipitate) + 3H⁺

Such precipitation due to hydrolysis does not take place in acidic media (because acid is like adding H+ to the right-hand side), and also, the precipitated matter can be redissolved in acid. This is why the solution goes nearly clear [Fe(III) in solution is a very pale yellow]. The next time, try mildly acidifying the solution FIRST before the addition of the iron (III) salt. You will obtain a rather clear solution.

Boiling or the addition of base hasten the hydrolysis (formation of precipitate).