Search engine optimization (SEO) :
Search engine optimization (SEO) is the process of improving the volume and quality of traffic to a web site from search engines. Usually, the earlier a site is presented in the search results, or the higher it "ranks," the more searchers will visit that site. The leading search engines, Google, Yahoo! and Microsoft, use crawlers to find pages for their algorithmic search results.

Search Engine :
A Web search engine is a search engine designed to search for information on the World Wide Web. Information may consist of web pages, images and other types of files. Some search engines also mine data available in newsbooks, databases, or open directories. Unlike Web directories, which are maintained by human editors, search engines operate algorithmically or are a mixture of algorithmic and human input.

Basically, a search engine is a software program that searches for sites based on the words that you designate as search terms. Search engines look through their own databases of information in order to find what it is that you are looking for.

Search engine optimization (SEO) is the process of improving the volume and quality of traffic to a web site from search engines via "natural" ("organic" or "algorithmic") search results for targeted keywords. Usually, the earlier a site is presented in the Search Engine Results Pages (SERPS) or the higher it "ranks", the more searchers will visit that site. SEO can also target different kinds of searches, including image search, local search, and industry-specific vertical search engines.

As a marketing strategy for increasing a site’s relevance, SEO considers how search algorithms work and what people search for. SEO efforts may involve a site’s coding, presentation, and structure, as well as fixing problems that could prevent search engine indexing programs from fully spidering a site. Another class of techniques, known as black hat SEO or spamdexing, use methods such as link farms and keyword stuffing that tend to harm search engine user experience. Search engines look for sites that employ these techniques and may remove them from their indices.

The initialism "SEO" can also refer to "search engine optimizers", terms adopted by an industry of consultants who carry out optimization projects on behalf of clients, and by employees who perform SEO services in-house. Search engine optimizers may offer SEO as a stand-alone service or as a part of a broader marketing campaign. Because effective SEO may require changes to the HTML source code of a site, SEO tactics may be incorporated into web site development and design. The term "search engine friendly" may be used to describe web site designs, menus, content management systems, URLs, and shopping carts that are easy to optimize.

search engine is the engine the search results (web sites) appear on, example, search Google for Manhattan internet marketing, and the first one that comes up is advernation.com , seo is the search engine optimization done to the web site in question (advernation.com)

"Search engine" is actually a general term for any internet portal that has World Wide Web search capabilities while SEO(Search engine Optimization) is the method of developing or improving the volume and quality of traffic to a web site from search engines via organic search results for targeted keywords.

Search engines is a directory listing that contains almost all the websites around the world, this is the place if your looking just about anything while search engine optimization is a process of optimizing websites to get more traffic in your site and the main goal of the SEO is to get your website rank higher in search engine’s and if your website is in the top listings of search engines, you’ll probably have more visitors on your site.