Amazon Redshift is a standard SQL database (based on Postgres) with MPP features that allow it to scale. These features also require you to conform your data model somewhat to get the best performance. It supports a large amount of the SQL standard and most tools that can speak to Postgres can use it unchanged.

BigQuery is not a database, in the sense that there it doesn't use standard SQL and doesn't provide JDBC/ODBC connectivity. It's a unique service with it's own API and interfaces. It provides limited support for SQL queries but most users interact with via custom code (Java, Python, etc.). Some 3rd party tools have added support for BigQuery but existing tools will not work without modification.

tl;dr - Redshift is better for interacting with existing tools and using complex SQL. BigQuery is better for custom coded interactions and teams who dislike SQL.

UPDATE 2017-04-17 - Here's a much more up to date summary of the cost and speed differences (wrapped in a sales pitch so YMMV). TL;DR - Redshift is usually faster and will be cheaper if you query the data somewhat regularly. http://blog.panoply.io/a-full-comparison-of-redshift-and-bigquery

UPDATE - Since I keep getting down votes on this () here's an up-to-date response to the items in the other answer:

Sizing your cluster:

• Redshift allows you to tailor your costs to your usage. If you want the fastest possible queries choose SSD nodes and if you want the lowest possible cost per GB choose HDD nodes. Start small and add nodes whenever you want.

Hourly costs when doing nothing:

• Redshift keeps your cluster ready for queries, can respond in milliseconds (result cache) and it provides a simple, predictable monthly bill.
• For example, even if some script accidentally runs 10,000 giant queries over the weekend your Redshift bill will not increase at all.

Speed of queries:

• Redshift performance is absolutely best in class and gets faster all the time. 3-5x faster in the last 6 months.

Indexing:

• Redshift has no indexes. It allows you to define sort keys to optimize performance from fast to insanely fast.

Vacuuming:

• Redshift now automatically runs routine maintenance such as ANALYZE and VACUUM DELETE when your cluster has free resource.

Data partitioning and distributing:

• Redshift never requires distribution. It allows you to define distribution keys which can make even huge joins very fast.
• {Ask competitors about join performance…}

Streaming live data:

• Redshift has 2 choices
  • Stream real time data into Redshift using Amazon Kinesis Firehose.
  • Skip ingestion altogether by querying your real time instantly on S3 as soon as it land (and at high speeds) using Redshift Spectrum external tables.

Growing your cluster:

• Redshift can elastically resize most clusters in a few minutes.

Multi zone:

• Redshift seamlessly replaces any failed hardware and continuously backs up your data, including across regions if desired.