What differences exist between the pay and free version of MSSQL2008R2?
Microsoft SQL Server 2008 R2
http://www.microsoft.com/sqlserver/2008/en/us/R2.aspx
and Microsoft SQL Server 2008 R2 Express
http://www.microsoft.com/express/Database/
A complete answer might explain why someone would choose one over the other, what the biggest annoyances are with going Express, non-intuitive things to look out for, etc.

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5 Answers
This page has a good comparison... The Express version is free, but it's limited in areas.
http://www.microsoft.com/sqlserver/2008/en/us/editions-compare.aspx

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Executive summary (Commonly missed features for the person running a small database):
SQL 2008 R2 Express edition is lacking:
SQL Server Agent (for scheduling jobs, including SQL replication)
support for DBs >10GB
Integration services
There are many other differences, detailed at the URL Moose posted.
SQL Express CAN be accessed from the network, but it is not enabled by default. A quick google will show how

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We often see shortened list of differences between Express and Standard. In my opinion, when you are to the point of considering this for a "production" environment deployment, you need to fully be aware of differences. Here's the full detailed list of differences between 2008R2 Standard and 2008R2 Express from msdn:
Scalability and Performance
- Number of CPUs ( 4 v.s. 1 )
- Maximum memory utilized ( 64GB v.s. 1GB )
- Maximum database size ( 524PB v.s. 10 GB )
- Direct query of index views using NOEXPAND hint
- Automatic indexed view maintenance
High Availability
- Log shipping
- Database mirroring
- Automatic corruption recovery from mirror
- Log stream compression
- Number of Failover clustering nodes ( 2 v.s. 0 )
- Backup compression
Virtualization Support
- Guest failover clustering support for virtualization
- Hyper-V live migration
Replication
- Publishing data from SQL Server to non SQL Server subscribers
Single Instance RDBMS Management
- Dedicated admin connection ( yes v.s. under trace flag )
- Policy automation (check on schedule and change)
- Maintenance plans
- Database mail
- Performance data collector
- System Center Operations Manager Management Pack
- Plan Guides
- User instances ( only for Express )
Application and Multi-Instance Management
- Able to enroll for multi-instance management.
Management Tools
- SQL Server Management Studio ( Standard v.s. Express)
- SQL Server Agent
- Database Engine Tuning Advisor
- SQL Server Profiler
Development Tools
- IntelliSense (Transact-SQL and MDX)
- Business Intelligence Development Studio
- SQL query, edit and design tools
- Version control support
- MDX edit, debug, and design tools
Programmability
- Full-text search
- Specification of language in query
- Service Broker (messaging) (client only for Express)
- XML/A support
- Web services (HTTP/SOAP endpoints)
- T-SQL endpoints
Spatial and Location Services
- Spatial results tab within SQL Server Management Studio
Complex Event Processing (StreamInsight)
- StreamInsight
Integration Services
- Integration Services designer and runtime
- Basic tasks and transformations in addition to those used by the Import and Export Wizard
- Log providers and logging
- Basic data profiling tools
- Programmable object model for extensibility
Data Warehouse
- Auto-generate staging and data warehouse schema
Analysis Services
- SQL Server Analysis Services backup
- Dimension, attribute relationship, aggregate, and cube design
- Translations
- Personalization extensions
Analysis Services-Advanced Analytic Functions
- Binary and compressed XML transport
Data Mining
- Comprehensive set of data mining algorithms
- Integrated data mining tools: wizards, editors, model viewers, query builder
Reporting
- Reporting Services memory limits ( 4Gb for Express )
- Model support
- Model Item Security
- Infinite click-through
- E-mail and file share subscriptions and scheduling
- Report history, executing snapshots, and caching
- SharePoint integration
- Shared component library
- Remote and non-relational data source support
- Data source, delivery, and rendering extensibility
- Report definition customization extension (RDCE)
Business Intelligence Clients
- Report Builder 3.0
- Excel 2007 and Visio 2007 Add-in support

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The most prominant difference is that SQL Express can only run on 1 of your systems processors rather than multi-proc. Personally, in my DEV environments I really can't tell the difference in performance.

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On the subject of hardware limitations, express is also limited to 1GB of RAM. – Shane Sep 01 '10 at 19:25
A few big things come to mind, you don't have an SQL Agent with express, and being a Publisher for replication isn't possible. (Because of not having an agent) And you are limited to the size of your DB.

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