This is an example of a tool where there is a massive shift in performance as you move from budget to high-end, to the extent that a high-end tool is to all intents and purposes a different tool to the low-end. It makes answering the question “why would I want one?” a bit tricky since the range of things you might do with a good one is much wider that those you would contemplate for a poor one. Hence it is simpler to treat these as two separate types of tool:
Ideal for cutting curved lines, (indeed without practice, that may be the only type you can cut!). If you need to cut out shapes, e.g. hole for a sink in a worktop, or make some ornate woodwork this may be the tool for the job. If you need a jigsaw then there are few alternatives - there are some jobs that only a jigsaw will do. The speed of cut is relatively slow, though ones with pendulum action will cut faster (and rougher). The tools are pretty small and light, and are often uncomfortable to use since you get a fair bit of vibration. They are not suited to being a general purpose saw (a circular saw will often be a better choice). The quality of the cut is moderate, and will need a fair amount of sanding etc., prior to finishing if it is to be on display.
This will do all of the things the budget one will do. However, it is a far more general purpose tool. It cuts quickly and smoothly with little or no vibration. It is much better at cutting straight lines, and can often be used with a straight edge or rip fence without the blade wandering to “interesting” angles. Try that with budget one and you will end up with a slanted cut or a broken blade.
Tool-less blade change is a given, as is a good speed controller. The base plate will be a solid cast-metal rather than a flexible pressed-steel one. With a fine or medium blade it will also give a very fine finish to a cut.
This fine finish quality will also open up the possibility of the production of patterns and jigs for use with routers and other tools.
You can also expect a much greater rigidity of design overall. With this comes better accuracy of alignment of parts, and an accurate blade support. Attention to detail like dust blowers and an integral light can make following your pattern much simpler. Soft start, and feedback speed control make for a more civilised user experience and also better quality of cut under load. Inclusion of soft shoe covers (to save marking the surface of the wood), rip fence or beam trammel attachment bush, and a motor rated for endurance and continuous use can also be expected.
Jigsaws are one of the less dangerous power tools, although can still give you a very nasty cut if not treated with respect. Unless using for long periods or cutting some thin sheet materials then noise level is usually low enough to not require ear defenders. Eye protection however is useful. The sawdust produced will usually be coarse and not too much of a inhalation risk. However repertory protection should be worn when cutting materials prone to making very fine dust like MDF.
Great care should be taken to keep your fingers out of the line of cut - especially when they are under the material and out of site!. Care should also be taken with the flex, and the use of a RCD protected supply is strongly recommended.
Features worth having include tool-less blade change (sometimes called SDS, just to confuse), a dust blower that keeps the cutting line clear of sawdust is very useful, as is a small light that illuminates the line you are cutting. A soft plastic sole plate shoe is handy in some cases since it can reduce marking of surfaces.
Jigsaw blades are simple looking things, but it is important to get ones that are made with decent materials and to a reasonable standard of construction. A blade stamped out of a sheet of steel will never cut a straight line due to having a burr on one side of it! So beware the bundle pack of 50 blades for five pounds.
Blades are available for wood, and metal cutting. Note that metal cutting blades can also be used on wood where a very slow and precise cut rate is required (e.g. scribe cutting internal corners on skirting). There are also specialist blades like grit edged ones. These are for cutting hard materials like tile, glass, and cast iron. Note that these will not replace dedicated tile or glass cutting machines, and are not ideal for cuts in glass or ceramics since jigsaws have no water cooling provision.
There are also down cut blades (for causing less tear out and chipping on finished surfaces - like laminate faced boards), and flush cut blades for cutting right up to a wall without the body of the saw getting in the way.
Check that you buy the correct blade shank for your tool. Most look similar, however there are some subtle variations and not all tools will take all blades.
Nothing special to look out for, or avoid, on safety grounds. If it requires a tool to change the blade, then check that the screw/hex heads of the blade clamp are still in a serviceable condition.