No you cannot use both plugins. Alidropship original plugin only supports 4 themes and has features (split by variation, currency switcher, ect) and less bugs(less CPU Usage, importing issues, ect) than the Alidropship Woo plugin. Though the original plugin is less customizable than the woo version.
The Alidropship woo plugin can use any theme supported by woocommerce (hundreds), and plugins (both woo and 3rd party) that can give a lot more customization and features to your website . Though you should be familiar with how woocommerce works to use this version of the plugin.
If you use one then decide to switch, you literally have to start your site over from scratch (total re-install of wordpress). So make your decision carefully. If you have zero experience in woocommerce or web design and development (optimizing server, site, speed, ect) you should use the original version. The themes supported are lighter, make fewer requests, use less CSS and Script (to minify) and basically have everything ready to go for you. But if you do have some experience, then go with the woo version and take more control over your site and how it will look and operate.
Considering your on the woo thread, you can assume all of us chose the woo version and for very good reason. With proper optimization we can create better looking and performing sites than the alidropship original can do (simply because the themes and features are very limited compared to woo). But most of use has had some experience with at least wordpress and woocommerce in the past or decided to do bootcamp style learning lol.