We do design websites for both big companies and small, and most businesses should avoid Wordpress. Discalimer: We try to avoid developing for Wordpress though if a client insists we can and have developed for them in Wordpress. Sometimes it is a good idea to use Wordpress, but make sure the below do NOT apply to you first. Though this is specific to Wordpress we do believe it applies to any or most CMS.

Security

Wordpress sites get hacked alot. As a business this is the worst thing to happen since many customers could lose trust in you as a company and become quite annoyed even if it isn't your fault. Why does Wordpress get hacked?

1. Because most sites are the same (and they use the same plugins) a hack found for one site will apply everywhere. That means a hacker will know instantly after he is done with one website whether he knows how to hack yours too. He might ignore odd websites and focus on simple prey.
2. Once a hack is found it is usually published with tutorials online. So now anyone can try to hack your website due to a Wordpress hacker.
3. Wordpress is Open Source, they tell people their security flaws, especially when they are fixed, but did you update your wordpress? Or do you still have those old security holes?
4. Anyone can write Themes/Plugins. Do you know what is behind the code? It looks nice, but maybe there is a major flaw. Maybe intentional or accidental. We trust most plugins, but one mistake can spell disaster for thousands or millions of sites, and it has before. There is SO MUCH rampant hacking.
5. It is easy. The code is all available online so people can look through and find flaws. Custom websites don't hand their code out for people to look through to find holes. It is much more annoying and usually means most people won't bother.

Security alone should tell businesses to AVOID Wordpress. This probably won't be an issue for you if you don't use plugins. Though it is possible for Wordpress itself to have a flaw that is hopefully more unlikely.

Speed

Wordpress is notoriously known (along with other CMS sites) as being slow. Sometimes unbearably slow, the average website seems to take 9 seconds. That is approximately 9x longer than what is usually acceptable. People might just avoid your business and click away if you don't load quickly enough. Your competitor might be open in another tab and ALREADY loaded. A slow website also appears un-professional. No one wants a customer annoyed with your store even if it isn't your fault. Why is Wordpress slow?

1. There is tons of code written so your life is simpler and websites are made without any programming knowledge. Thats great, except many times you just want to see your code not have a huge application running on your server. Some sites could have no server code at all, but with Wordpress that isn't possible, it needs to take time to process requests and get everything ready before it inserts lots of code and runs lots of plugins. If you change your website alot you might be willing to make it slower for each user so you can easily update your site. Keep in mind a custom website allows you to change what you specifically need to change, with a lot less overhead.
2. Themes & Plugins.... You trust anyone to write fast code? These pieces of code can be badly written, but the worst part? When I developed for Wordpress I saw many times plugins being run that were not even needed on the page!!! That means a voting plugin code might be downloaded and take an extra second for NO REASON. Imagine if you have 10 plugins, that can add up quickly!
3. Do you know how to install your Wordpress properly? Many times it can be made faster if you change settings, but this does take some knowledge and sometimes trusting more plugins. It might be doable for the average person, so if you use Wordpress, take a few hours to learn how to make it faster. Now test your whole website to make sure making it faster didn't break anything. Perhaps test step by step so you know what trick ruined your website. To be fair this applies to ALL websites, but usually Wordpress websites are in the hands of the business to maintain not your web designer, who doesn't care about making it fast. You already payed him and you think its fast because your the only one looking at it right now. Did they install a cache, do you want one? If your website doesn't change Cache lots of pages, if it does, then you need to find other tricks.
4. Wordpress was made for any website to run any way they want. A custom website is tailor made, so they can make sure the code runs fast since it was designed for you, Wordpress can't. They handle every case and speed is a lower priority than making sure every type of website will work.

The Look of Your Website

Wordpress and most CMS websites do not look professional. Why?

1. Wordpress websites look the same as each other. This is bad because now as a customer I see a website that doesn't stand out, but worse I know it was made with Wordpress, what is meant for people to make cheap, simple websites. I wouldn't give a business much credit. Do I really trust them to spend money where it counts? They don't bother to make a nice website.... Does the website even work if I want to buy something, it's Wordpress? Would you have a storefront that looks dirty? People judge books by the cover, sad, but true.... Tip: Make sure that any design decisions are made before you discuss using Wordpress. If they won't transfer the look to Wordpress, don't bother. Wordpress doesn't need to look ugly.
2. Anyone can make a Wordpress site. That means you might not be getting a professional, so how will you look professional if your not using a professional? Tip: Make sure to see a portfolio! You might not be amazon, but don't settle for a website that looks 10 years old.

Cost & Efficiency

This probably doesn't apply to most businesses, suprisingly.

Do you expect lots of visitors? Wordpress is meant for blogging. If you have lots of visitors, and you are doing anything complicated, Wordpress won't scale. That means that it can't handle the traffic. PHP, the language that underlies Wordpress is difficult, but very feasible to scale if you know what your doing. A huge application that you will need to twist to allow tons of users will probably be impossible for most users. Expect to pay tons of money to do that. Either pay a lot of money to a web professional and invest originally in a website that will scale, or keep in your budget that you will need to create a new website when you have tons of visitors. Don't wait until it is too late and users are bringing down your website. Be proactive. If your website is moving slowly it might mean there is too much traffic, keep an eye on web traffic. Tip: This probably will affect few startups, who might be lured into Wordpress, major companies have enough money to avoid wordpress.

Who should use Wordpress? If you blog and it is your main selling point, than Wordpress might make sense. If you want to have a website as cheap as possible, and expect that people will come to your business regardless of a website, than I suppose you can use Wordpress. In today's age people will likely not bother even trying out your store if you don't have a professional website. It might be better to have nothing than something bad....