Advanced search: USASearch compared to Apache Solr Search implemented by Acquia
I didn't go too deep into USASearch, but it seems to me as a tool similar to Google Custom Search Engine.

So, let's classify them first. We'll call all such services as Search Providers. On the other hand, we'll call ApacheSOLR instances (whether through Acquia or otherwise), regular search index in Drupal or other such index hosting services as Indexes.

The general difference between Search Providers and Indexes is mostly in the level of control.

Search Providers


Quicker to setup for static websites as it is just HTML/JS code.
Ideal for simple static content with limited customization.
You don't have to worry about the index at all. That is your provider's job.


Indexes


For Drupal, it is just a matter of installing a module.
You get complete control over results, templating, filtering, facet search, etc...
You have to take care of indexing content, permissions, and search configuration.


Generally speaking, go for Indexes, or in your case, Acquia Search. The configuration and indexing is one time (and sometimes not even that) and the added benefits are huge. There are very few scenarios where I would pick a search provider instead.