The first thing you need to know is that the two options are not exclusive one to another, they are actually complementary and can both be used in a successful marketing campaign on the internet.
Each method has its advantages and disadvantages and deciding to use one over another should be done based on the expected results.
Google AdWords is an online advertising service offered by Google that allows you to display a link to your website when certain words (or keywords) are searched on Google or when they are present on one of the websites that is part of their marketing network. AdWords marketing campaigns can be run at local level, national or international.
Optimizing your website (SEO) it's a set of actions that have as effect a better ranking of your website in search engines. By using SEO, the website will be better indexed by the search engines and this usually results in naturally allowing the website to show up in searches on top of other websites that have the same activity or market.
Who should use them?
Google Adwords is addressed to companies that would like to have a result on a short term, as quickly as just hours after the Google Adword campaign is launched. Using Google Adwords will increase your website traffic for a short period of time or more precisely usually only for the period of time the campaign is run. Very often, once the Adword campaign stops, the web traffic will drop as well.
Search Engine Optimization for Google is addressed to companies that wish to have a lasting presence on Internet, and would like their investment to have an effect for a long period of time.
As any good thing takes time, the SEO effects will be visible usually after a couple of weeks. Usually it is not a good idea to rush and do all the changes very quickly, because search engines will be able to detect the campaign and might slow the grow of a website in ranks. To be on the safe side, we usually say it takes up to two months for the effects to be visible, but it really depends on the search keys that are chosen to have the website optimized for and also it depends a lot on the competition present on the market.
Optimizing your website for search engine indexing has the benefit that it offers lasting effects long after the campaign is stopped.
Google Adwords is based on advertising ads that are presented in Google Adword network (both Google Search and associated websites). Google ads will be presented when an Internet user chooses certain words in his search, or when a website in the associated network has certain words on its page. For example it is more likely a web-page reader will be interested in buying insurance if he is reading about cars on a website. Also if you are looking for an artist or a song on Google, it is quite possible you are interested in buying his last album or maybe his greatest hits.
Once the user clicks the the link, your page will be opened. It is quite possible you will be able to transform this visitor into a payer customer. Unfortunately users have a tendency to ignore advertising run like this, and despite any form of advertising that “pops in your face” while you are reading or accessing a website.
Search Engine Optimization has the purpose to prepare your website for better indexing by the Search Engine. Indexing is done based on your website content, and when the website is prepared to highlight certain words (keys), it will offer a better rank in searches that are looking for these keys. This offers a long lasting effect in search ranks, and offers a very good visibility for your website, your services and /or products.
Costs
When using Google Adwords there is a cost based on a PPC (Pay-Per-Click) or based on the number of displays. Your cost will depend on the number of displays and on the number of visitors that click on your ad. This ads need to be properly configured, as they can become very expensive very quickly if they are displayed too often and to unnecessary users.
Using SEO services implies higher costs initially, but the return of investment will also be higher, because of the long lasting effect of such a campaign. Another benefit is that you can always adjust the keys you would like to have your website indexed against.
Google Adwords will be displayed in the sponsored area of Google Search. This can be on top of all “natural” results or on the right side of the screen. This advertisements will be published usually on a different font and/or on a different background colour. Also they can be displayed on Google's advertising network that is composed of big websites like Google Maps, Google Gmail, Amazon.com , NYTimes, howstuffworks or business.com, but also they can be displayed on small websites like blogs, video websites (youtube.com) or personal websites.
When using SEO services, the website will be ranking better in searches. "Normal" websites will be displayed on the left side and their order is based on the rankings calculated by the search engine. Research has shown that 80% of users that are using a search engine prefer the websites that are listed in the search engine because of organic (natural) ranking.
As you've probably seen already, both methods work based on your page being present on the Google Search website. Also both methods work based on indexing keys (words).
Success of both methods will highly depend on the search keys that are used:
You have the possibility to use both methods, so you could have the best of the two worlds: both a high return on the long time, and also a very good return and presence on the Internet in the first weeks when SEO is not effective just yet.