Image credit: WPCentral
Using these non-public APIs in a third party app is a clear violation of Instagram's terms of service. The app almost certainly breaks section 10 of Instagram's terms of service ("you must not access Instagram's private API by means other than those permitted by Instagram") and will likely also in violation of section 2 of Instagram API's terms of sevice ("you shall not use Instagram APIs for any application that replicates or attempts to replace the essential user experience of Instagram.com or the Instagram iPhone App"). Theoretically, this could lead to the suspension or deletion of an account that uses such an app, although in practise this is unlikely.
It's more likely that Instagram would seek to block such an app. It could do this by changing the API, but this approach is relatively unlikely, given the impact it would have on existing installations of the official app on other platforms. However, there are other avenues open to Instagram, including legal action, and traditionally Instagram has been aggressive in maintaining a policy of only allowing the official apps write access to the service. The most likely scenario? Instagram will ask Microsoft to pull the app from the Windows Phone Store.
This means you would be using the app at your own risk and there's no guarantee it will available in the long term. However, provided you are aware of the situation before purchasing the app, there's no real cause for complaint, at least from a consumer viewpoint.
It's worth noting that another third party Instagram client, Instagraph, also allows you to uploads photos to the service. When we first heard about the app we were doubtful it would get approved by Microsoft or long survive Instagram's attention, but it seems we were to pessimistic as the app is now available in the Windows Phone Store and juding by the reviews has a number of happy users.
Instagraph does arguably fall into a slightly different category to Itsdagram because it is using a server side work around. This means it technically may not be in breach of the letter of the Instagram API terms and conditions, although this judgement is open to debate, and it is fair to say that it is certainly breaking the spirit of Instagram's terms and conditions.
And what about an official Instagram app? Well, some rumours suggest that it might not be far off. Unconfirmed sources, claiming to be close to Microsoft, suggest that an app has been developed, but that it is subject to approval by Instagram. Nokia's upcoming event on May 14th might be worth keeping an eye on, but for now there's still no official world on the arrival on an official Instagram app, leaving plenty of room for third party alternatives.