Some of the best features of ArcGIS (32bit flavor - not ArcGIS Pro) are in the cartography and vector editing categories. ArcGIS has good tools to produce quality maps. Some tools include the use of ESRI's "data driven pages", which allow users to tile maps on small pages for large areas with high detail (large scale).
Another nice feature in ArcGIS (only available at the Standard and Advanced license level) is vector editing with topology within a Geodatabase. The topology tools work well and have good tools built into them to allow the user to find topological errors and fix them. Vector topology is an essential tool for ensuring data integrity.
The costs of this software are enormous for the initial purchase price and the subsequent yearly fees for the product. Not having recent quotes, but in the past pricing for Standard and Advanced licensing were $7,000 and $12,000 respectively for initial purchase. Then yearly fees are incurred to get "updates" to the software and tech support. With $1500 and $3000 yearly maintenance fees respectively for Standard and Advanced, the costs per year can rapidly escalate when multiple licenses are purchased. The annual fees are practically mandatory to stay current with the community and client needs.
Proprietary data format of the Geodatabase. The ESRI geodatabase can only be created and written to by ESRI software (Standard and Advanced licenses). This design creates vendor lock-in, such that organizations need ESRI software to access and edit geodatabases. This design enforces sales of the ArcMap software, as ESRI geodatabases demand ESRI software to be fully utilized. This also creates an environment where a few special operators with special software can access the data. This discourages the accessibility and sharing of GIS data information. Being that GIS data is information, and information should be shared, the proprietary data format is an obstacle to creating easily accessible information within an organization and with other organizations.
The performance of ArcMap (32bit) is poor relative to today's computing standards. 32 bit architecture is very outdated and when performing spatial operations, the delay in processing costs users time. Since ArcMap came out and was re-written from scratch in the early 2000s, there have been continual issues with bugs and performance. Fixes to the bugs are slow to be resolved. Sometimes the application hangs up (not responding) for seemingly basic tasks.
Lack of modern algorithms. Example would be the hydrologic routing tools that are from the early 1990s and have not been updated with current modeling features such as Multi-Flow Direction (MFD) modeling of where water flows, and how much accumulation.
Tech support is a paid service along with updates to the software through yearly annual fees. The first level tech support has often left a disappointing feeling, where either the topic was being read directly from the help menu, or the technician kept trying to solve the problem with educated guesses, rather than knowing what the solution is. A substantial amount of time has been lost with tech support at the tier 1 level. The tier 2 level of support is good. The technicians know the products more specifically, and are generally more helpful to resolve the issue.
Some spatial functions are only available in the top tier product (such as identity). This aspect was really frustrating as a user and for an organization. The difference between the Standard and Advanced licensing is about $5,000 (last checked). The interesting thing is that the Basic, Standard and Advanced licenses are all the same product, just with varying levels of provided tools. One such tool is the identity function. This is a commonly overlay function in any GIS, however, ArcMap only provides this simple overlay at the advanced level.
Lack of general interoperability. Over the years, open data standards such as Web Mapping Services (WMS) and Web Feature Services (WFS) have provided open formats to work with GIS data. WFS is designed to allow editing of features over a web protocol, but in order to implement this in ArcMap, $10,000 software needs purchased (ArcGIS for Server). There is currently no way to write to an open database GIS format from ESRI software, without purchasing additional software.
Be careful. ESRI has managed to gain major market share and monopolized the GIS software. Over the years, other open data formats have nudged ESRI to move to more supported open data formats.
Evaluate your needs. Check to see what your organization needs GIS services for. Quantum GIS (QGIS) is a great open source desktop software that meets the geospatial communities needs, with hundreds if not thousands of user contributed add-ons and tools. Some cases may require the purchase of an ESRI license. Explore other options before falling into the vendor lock-in trap. Open source software in the geospatial world, is really advanced and amazing. Consult with a user of GIS in the FOSS world, to gain information about how different products can meet the needs of an ArcMap user.
Setting up a GIS Data Library as a PostGIS/PostgreSQL server was a challenge to integrate with ESRI. The good news is that SDE is a deprecated product and the libraries are now included with ArcMap. This allows ArcMap to access Database servers and PostGIS data formats as read-only. This is ideal for setting up large, state-wide layers of GIS data, with ArcMap able to connect and clip or select features from it, and place the data in the local Geodatabase for editing.